| L(s) = 1 | + 26·7-s − 1.38e3·13-s + 1.54e4·19-s − 1.17e5·25-s + 2.33e5·31-s − 6.02e5·37-s + 1.33e6·43-s − 1.64e6·49-s − 4.44e6·61-s + 2.60e6·67-s − 3.67e6·73-s + 7.78e6·79-s − 3.59e4·91-s + 1.69e6·97-s − 3.19e7·103-s + 2.53e7·109-s + 2.63e7·121-s + ⋯ |
| L(s) = 1 | + 0.0286·7-s − 0.174·13-s + 0.516·19-s − 1.50·25-s + 1.40·31-s − 1.95·37-s + 2.56·43-s − 1.99·49-s − 2.50·61-s + 1.05·67-s − 1.10·73-s + 1.77·79-s − 0.00499·91-s + 0.188·97-s − 2.87·103-s + 1.87·109-s + 1.35·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.040565735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.040565735\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 + 23474 p T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 13 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 26382938 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 691 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 153457666 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7723 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 5851297774 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 29682365098 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 116572 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 301213 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 287975138482 T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 667720 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 892674656446 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1124041475194 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3498623125318 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2222401 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 1300183 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 10325701752338 T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 1836631 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3890947 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 18647411627254 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 87379168132978 T^{2} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 845573 T + p^{7} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.11739413408756459705299771713, −9.789180310077613589220570625457, −9.287079953463661138719436470326, −9.003061272644254295338813078445, −8.251516916846413747528589416114, −7.978882733509101567832951353503, −7.49300300608766292077548941160, −7.07443946668524768231340691731, −6.25669110557524382290624424460, −6.23964360007870685462059804919, −5.36044590057905573361207178058, −5.13880394716949207424861827168, −4.25195358498139719732233113786, −4.13335589500271499912832775157, −3.10641081497515060798683434136, −3.02863143742613369192777752196, −2.02418463893067990754998817126, −1.71559272334365003825109513924, −0.907836532840156117108549705649, −0.33042568781132899867199370866,
0.33042568781132899867199370866, 0.907836532840156117108549705649, 1.71559272334365003825109513924, 2.02418463893067990754998817126, 3.02863143742613369192777752196, 3.10641081497515060798683434136, 4.13335589500271499912832775157, 4.25195358498139719732233113786, 5.13880394716949207424861827168, 5.36044590057905573361207178058, 6.23964360007870685462059804919, 6.25669110557524382290624424460, 7.07443946668524768231340691731, 7.49300300608766292077548941160, 7.978882733509101567832951353503, 8.251516916846413747528589416114, 9.003061272644254295338813078445, 9.287079953463661138719436470326, 9.789180310077613589220570625457, 10.11739413408756459705299771713
