| L(s) = 1 | − 96·7-s − 200·17-s + 2.33e3·23-s + 7.02e3·25-s + 1.29e4·31-s + 4.56e3·41-s − 5.47e4·47-s − 2.40e4·49-s + 2.06e5·71-s + 3.99e4·73-s + 2.47e5·79-s + 8.46e4·89-s − 9.95e4·97-s − 1.80e5·103-s − 3.03e5·113-s + 1.92e4·119-s + 2.96e5·121-s + ⋯ |
| L(s) = 1 | − 0.740·7-s − 0.167·17-s + 0.920·23-s + 2.24·25-s + 2.41·31-s + 0.424·41-s − 3.61·47-s − 1.43·49-s + 4.86·71-s + 0.877·73-s + 4.46·79-s + 1.13·89-s − 1.07·97-s − 1.67·103-s − 2.23·113-s + 0.124·119-s + 1.84·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.371651595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.371651595\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 7028 T^{2} + 24404246 T^{4} - 7028 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 48 T + 15502 T^{2} + 48 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 296436 T^{2} + 49128544726 T^{4} - 296436 p^{10} T^{6} + p^{20} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 894228 T^{2} + 396323515894 T^{4} - 894228 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 100 T + 2767462 T^{2} + 100 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 6794580 T^{2} + 21506967947254 T^{4} - 6794580 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 1168 T + 10055470 T^{2} - 1168 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 31255380 T^{2} + 976386653995702 T^{4} - 31255380 p^{10} T^{6} + p^{20} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 6464 T + 65013054 T^{2} - 6464 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 241262580 T^{2} + 24149916431784598 T^{4} - 241262580 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2284 T + 146603254 T^{2} - 2284 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 10845276 p T^{2} + 96250708269010006 T^{4} - 10845276 p^{11} T^{6} + p^{20} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 27360 T + 591338206 T^{2} + 27360 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 1039152180 T^{2} + 595616955270391126 T^{4} - 1039152180 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 1537424180 T^{2} + 1399694789142612374 T^{4} - 1537424180 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 741098540 T^{2} + 1478044222094100534 T^{4} + 741098540 p^{10} T^{6} + p^{20} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 1366835860 T^{2} + 3274116308996825526 T^{4} - 1366835860 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 103344 T + 6217736974 T^{2} - 103344 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 19988 T + 2541602870 T^{2} - 19988 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 123936 T + 9855929374 T^{2} - 123936 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 10047855188 T^{2} + 55381071937674414326 T^{4} - 10047855188 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 42316 T + 4292401174 T^{2} - 42316 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 49788 T + 16391371462 T^{2} + 49788 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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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
−8.022183387557427283346709360864, −7.38972688672310205566512998695, −7.04807935144683120879317297867, −6.71126608463499125381783007705, −6.62526604134952859068826132782, −6.57683677947969463346714252115, −6.18184632296640829864592583885, −6.17421100959674518707073979630, −5.26649295916102066225488697585, −5.26305641055301408736743077913, −5.03430027716428851880899563511, −4.74178007411790854998854416100, −4.65525050074578693126863890416, −4.07638563962504100767418256542, −3.59512446852714993596139320887, −3.52699034977226172506393414421, −3.23027113170659459449867045506, −2.73801624779992579458299459931, −2.63450772598622689659552819522, −2.28081684856554294278192896635, −1.59980091469193879661132269133, −1.50303773501428859636856697214, −0.811138877794429212437438204470, −0.54543966784961438716635172697, −0.53165674816913544331693987421,
0.53165674816913544331693987421, 0.54543966784961438716635172697, 0.811138877794429212437438204470, 1.50303773501428859636856697214, 1.59980091469193879661132269133, 2.28081684856554294278192896635, 2.63450772598622689659552819522, 2.73801624779992579458299459931, 3.23027113170659459449867045506, 3.52699034977226172506393414421, 3.59512446852714993596139320887, 4.07638563962504100767418256542, 4.65525050074578693126863890416, 4.74178007411790854998854416100, 5.03430027716428851880899563511, 5.26305641055301408736743077913, 5.26649295916102066225488697585, 6.17421100959674518707073979630, 6.18184632296640829864592583885, 6.57683677947969463346714252115, 6.62526604134952859068826132782, 6.71126608463499125381783007705, 7.04807935144683120879317297867, 7.38972688672310205566512998695, 8.022183387557427283346709360864
