| L(s) = 1 | − 4·5-s − 20·11-s + 8·13-s − 24·17-s + 44·19-s + 72·23-s + 125·25-s + 76·29-s + 184·31-s + 30·37-s − 432·41-s − 328·43-s − 520·47-s − 146·53-s + 80·55-s − 460·59-s + 628·61-s − 32·65-s − 556·67-s − 1.18e3·71-s + 1.02e3·73-s + 104·79-s − 648·83-s + 96·85-s − 896·89-s − 176·95-s + 1.84e3·97-s + ⋯ |
| L(s) = 1 | − 0.357·5-s − 0.548·11-s + 0.170·13-s − 0.342·17-s + 0.531·19-s + 0.652·23-s + 25-s + 0.486·29-s + 1.06·31-s + 0.133·37-s − 1.64·41-s − 1.16·43-s − 1.61·47-s − 0.378·53-s + 0.196·55-s − 1.01·59-s + 1.31·61-s − 0.0610·65-s − 1.01·67-s − 1.97·71-s + 1.64·73-s + 0.148·79-s − 0.856·83-s + 0.122·85-s − 1.06·89-s − 0.190·95-s + 1.92·97-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
| \(L(2)\) |
\(\approx\) |
\(0.6423873395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6423873395\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T - 109 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T - 931 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 24 T - 4337 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 44 T - 4923 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 72 T - 6983 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 38 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 184 T + 4065 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 30 T - 49753 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 216 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 164 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 520 T + 166577 T^{2} + 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 146 T - 127561 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 460 T + 6221 T^{2} + 460 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 628 T + 167403 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 556 T + 8373 T^{2} + 556 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 592 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1024 T + 659559 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 104 T - 482223 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 324 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 896 T + 97847 T^{2} + 896 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 920 T + p^{3} T^{2} )^{2} \) |
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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
−9.146795587603413720654597412551, −8.608631575417185464697964357267, −8.406775288103347904102964578023, −7.934811107406342843475578495141, −7.71342670613578346275128236124, −6.96639465165704770090151683921, −6.86419916446445564615308252137, −6.42678431701514713838670882063, −6.00289942750689493643912683824, −5.29562113833009318241056819934, −5.07784735614588669937628326463, −4.63870743234786396276210762386, −4.32135133891677080351519333796, −3.39561290030609142901828588870, −3.33829678064035108361884927186, −2.77629112207414538387656546430, −2.23156081553992269578077556720, −1.41719066393844965239040771789, −1.11176387415633911350809346862, −0.18302262650158971970537391434,
0.18302262650158971970537391434, 1.11176387415633911350809346862, 1.41719066393844965239040771789, 2.23156081553992269578077556720, 2.77629112207414538387656546430, 3.33829678064035108361884927186, 3.39561290030609142901828588870, 4.32135133891677080351519333796, 4.63870743234786396276210762386, 5.07784735614588669937628326463, 5.29562113833009318241056819934, 6.00289942750689493643912683824, 6.42678431701514713838670882063, 6.86419916446445564615308252137, 6.96639465165704770090151683921, 7.71342670613578346275128236124, 7.934811107406342843475578495141, 8.406775288103347904102964578023, 8.608631575417185464697964357267, 9.146795587603413720654597412551
