| L(s) = 1 | + 6·5-s + 36·11-s + 124·13-s + 114·17-s + 76·19-s − 24·23-s + 125·25-s − 108·29-s + 112·31-s + 178·37-s − 756·41-s − 344·43-s − 192·47-s − 402·53-s + 216·55-s + 396·59-s − 254·61-s + 744·65-s + 1.01e3·67-s − 1.68e3·71-s − 890·73-s − 80·79-s + 216·83-s + 684·85-s − 1.63e3·89-s + 456·95-s + 2.02e3·97-s + ⋯ |
| L(s) = 1 | + 0.536·5-s + 0.986·11-s + 2.64·13-s + 1.62·17-s + 0.917·19-s − 0.217·23-s + 25-s − 0.691·29-s + 0.648·31-s + 0.790·37-s − 2.87·41-s − 1.21·43-s − 0.595·47-s − 1.04·53-s + 0.529·55-s + 0.873·59-s − 0.533·61-s + 1.41·65-s + 1.84·67-s − 2.80·71-s − 1.42·73-s − 0.113·79-s + 0.285·83-s + 0.872·85-s − 1.95·89-s + 0.492·95-s + 2.11·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.441363931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.441363931\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T - 89 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 36 T - 35 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 62 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 114 T + 8083 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 p T - 3 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 24 T - 11591 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 112 T - 17247 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 178 T - 18969 T^{2} - 178 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 378 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 192 T - 66959 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 402 T + 12727 T^{2} + 402 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 396 T - 48563 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 254 T - 162465 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1012 T + 723381 T^{2} - 1012 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 840 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 890 T + 403083 T^{2} + 890 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 80 T - 486639 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 1638 T + 1978075 T^{2} + 1638 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1010 T + p^{3} T^{2} )^{2} \) |
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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
−8.949089067676813534953711212213, −8.721452343943278549383145304237, −8.441200011104557433124453276930, −8.152136421632625136733224169077, −7.47029990351427067753005347970, −7.17816029611407992122616799436, −6.61889124880372168107656230822, −6.30815119834359154151319725987, −5.82410703443406472023309587185, −5.78773001286347016533481159173, −5.01606429151151892555810103087, −4.74165644082363399631367881562, −4.02088769281063797096125660513, −3.53044786399619899466539462131, −3.24262779659524419082455593518, −3.02855194034514112733958211792, −1.75236900299777579815509179670, −1.61695183772888529540617440184, −1.12547451513669133892225638442, −0.59878550989175080128156856499,
0.59878550989175080128156856499, 1.12547451513669133892225638442, 1.61695183772888529540617440184, 1.75236900299777579815509179670, 3.02855194034514112733958211792, 3.24262779659524419082455593518, 3.53044786399619899466539462131, 4.02088769281063797096125660513, 4.74165644082363399631367881562, 5.01606429151151892555810103087, 5.78773001286347016533481159173, 5.82410703443406472023309587185, 6.30815119834359154151319725987, 6.61889124880372168107656230822, 7.17816029611407992122616799436, 7.47029990351427067753005347970, 8.152136421632625136733224169077, 8.441200011104557433124453276930, 8.721452343943278549383145304237, 8.949089067676813534953711212213
