| L(s) = 1 | + 15·4-s + 45·9-s + 22·11-s + 161·16-s + 170·19-s + 330·29-s − 166·31-s + 675·36-s − 956·41-s + 330·44-s + 605·49-s + 580·59-s + 514·61-s + 1.45e3·64-s − 626·71-s + 2.55e3·76-s − 1.66e3·79-s + 1.29e3·81-s − 50·89-s + 990·99-s − 596·101-s − 2.10e3·109-s + 4.95e3·116-s + 363·121-s − 2.49e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 15/8·4-s + 5/3·9-s + 0.603·11-s + 2.51·16-s + 2.05·19-s + 2.11·29-s − 0.961·31-s + 25/8·36-s − 3.64·41-s + 1.13·44-s + 1.76·49-s + 1.27·59-s + 1.07·61-s + 2.84·64-s − 1.04·71-s + 3.84·76-s − 2.36·79-s + 16/9·81-s − 0.0595·89-s + 1.00·99-s − 0.587·101-s − 1.84·109-s + 3.96·116-s + 3/11·121-s − 1.80·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.308755971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.308755971\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 5 p^{2} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 605 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4390 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9385 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 85 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 23850 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 165 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 83 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 101305 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 478 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 158950 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 191770 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 168735 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 290 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 257 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 650 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 313 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 35570 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 830 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 434610 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 25 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1357310 T^{2} + p^{6} T^{4} \) |
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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
−11.60694674555497992252736607939, −11.54213027214641152687927214476, −10.53662856777631530380748067127, −10.36964352960762086142845339206, −9.965016799708577400784328737871, −9.588995696701131413299045778697, −8.695656477237107199038474109618, −8.255711567505912528332529302173, −7.49347567826113177046821639357, −7.17451958427529985825368591497, −6.81734974531866005061112831124, −6.55599222930224853810673994425, −5.61384171330200522978598617712, −5.26144692797919341061226776947, −4.38610793530936366420319678697, −3.59994263003135050749452011140, −3.12993463699529367610036505116, −2.29680241248037722949244067840, −1.42635189578678313745218765382, −1.14357382268898568326301184639,
1.14357382268898568326301184639, 1.42635189578678313745218765382, 2.29680241248037722949244067840, 3.12993463699529367610036505116, 3.59994263003135050749452011140, 4.38610793530936366420319678697, 5.26144692797919341061226776947, 5.61384171330200522978598617712, 6.55599222930224853810673994425, 6.81734974531866005061112831124, 7.17451958427529985825368591497, 7.49347567826113177046821639357, 8.255711567505912528332529302173, 8.695656477237107199038474109618, 9.588995696701131413299045778697, 9.965016799708577400784328737871, 10.36964352960762086142845339206, 10.53662856777631530380748067127, 11.54213027214641152687927214476, 11.60694674555497992252736607939
