| L(s) = 1 | − 12·7-s + 16·11-s − 48·13-s + 24·17-s − 56·23-s + 80·31-s − 112·37-s − 56·41-s − 8·43-s − 16·47-s + 72·49-s + 120·53-s − 24·61-s − 8·67-s − 272·71-s − 108·73-s − 192·77-s − 9·81-s + 272·83-s + 576·91-s + 148·97-s + 152·101-s + 124·103-s − 160·107-s + 144·113-s − 288·119-s − 312·121-s + ⋯ |
| L(s) = 1 | − 1.71·7-s + 1.45·11-s − 3.69·13-s + 1.41·17-s − 2.43·23-s + 2.58·31-s − 3.02·37-s − 1.36·41-s − 0.186·43-s − 0.340·47-s + 1.46·49-s + 2.26·53-s − 0.393·61-s − 0.119·67-s − 3.83·71-s − 1.47·73-s − 2.49·77-s − 1/9·81-s + 3.27·83-s + 6.32·91-s + 1.52·97-s + 1.50·101-s + 1.20·103-s − 1.49·107-s + 1.27·113-s − 2.42·119-s − 2.57·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8737723198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8737723198\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 660 T^{3} + 6014 T^{4} + 660 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 8 T + 252 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 21360 T^{3} + 319874 T^{4} + 21360 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} + 5448 T^{3} - 163198 T^{4} + 5448 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1004 T^{2} + 509190 T^{4} - 1004 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 56 T + 1568 T^{2} + 45528 T^{3} + 1241282 T^{4} + 45528 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1424 T^{2} + 980610 T^{4} - 1424 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 40 T + 1722 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 112 T + 6272 T^{2} + 316848 T^{3} + 13874882 T^{4} + 316848 p^{2} T^{5} + 6272 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 28 T + 2382 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 3960 T^{3} - 5004286 T^{4} - 3960 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} - 33456 T^{3} - 9745438 T^{4} - 33456 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 120 T + 7200 T^{2} - 409080 T^{3} + 22882562 T^{4} - 409080 p^{2} T^{5} + 7200 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8600 T^{2} + 42555378 T^{4} - 8600 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 4574 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 75000 T^{3} - 16429246 T^{4} - 75000 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 136 T + 14322 T^{2} + 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 108 T + 5832 T^{2} + 214596 T^{3} - 3272626 T^{4} + 214596 p^{2} T^{5} + 5832 p^{4} T^{6} + 108 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 556 T^{2} + 67139430 T^{4} + 556 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 272 T + 36992 T^{2} - 3994320 T^{3} + 370520834 T^{4} - 3994320 p^{2} T^{5} + 36992 p^{4} T^{6} - 272 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 23804 T^{2} + 265665990 T^{4} - 23804 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 74 T + 2738 T^{2} - 74 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.80974478801260236758729488396, −6.73288979876455073823156135383, −6.26705047671334290082017060210, −6.08887138132199645683454806145, −5.98542277567771656410914349209, −5.62965163750898561125636311123, −5.60269217299573722245644836075, −5.06699503737823611060079944460, −4.86276939573952778361793416613, −4.81615609996670574802233513871, −4.58333179945591549956134926097, −4.15028956717892905626598178127, −4.05552546070067765369016699479, −3.58204783091550739987964334013, −3.43134904005832387549738829898, −3.32135062171533998260440095972, −2.97813784024939668658947059633, −2.57274028360478869410010828879, −2.47385943338585087728238331087, −2.08558694775451632283805947882, −1.84710467920906373025609145796, −1.42424388260694442919702039601, −1.02687571808883546921744509434, −0.33663142202162034631448183360, −0.27381233776609975097632334105,
0.27381233776609975097632334105, 0.33663142202162034631448183360, 1.02687571808883546921744509434, 1.42424388260694442919702039601, 1.84710467920906373025609145796, 2.08558694775451632283805947882, 2.47385943338585087728238331087, 2.57274028360478869410010828879, 2.97813784024939668658947059633, 3.32135062171533998260440095972, 3.43134904005832387549738829898, 3.58204783091550739987964334013, 4.05552546070067765369016699479, 4.15028956717892905626598178127, 4.58333179945591549956134926097, 4.81615609996670574802233513871, 4.86276939573952778361793416613, 5.06699503737823611060079944460, 5.60269217299573722245644836075, 5.62965163750898561125636311123, 5.98542277567771656410914349209, 6.08887138132199645683454806145, 6.26705047671334290082017060210, 6.73288979876455073823156135383, 6.80974478801260236758729488396
