| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s + 16-s − 5·19-s + 6·22-s + 8·25-s − 32-s + 5·38-s + 15·41-s + 7·43-s − 6·44-s + 5·49-s − 8·50-s − 3·59-s + 64-s + 10·67-s − 5·73-s − 5·76-s − 15·82-s + 21·83-s − 7·86-s + 6·88-s + 18·89-s + 16·97-s − 5·98-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 1/4·16-s − 1.14·19-s + 1.27·22-s + 8/5·25-s − 0.176·32-s + 0.811·38-s + 2.34·41-s + 1.06·43-s − 0.904·44-s + 5/7·49-s − 1.13·50-s − 0.390·59-s + 1/8·64-s + 1.22·67-s − 0.585·73-s − 0.573·76-s − 1.65·82-s + 2.30·83-s − 0.754·86-s + 0.639·88-s + 1.90·89-s + 1.62·97-s − 0.505·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8694564229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8694564229\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.458409839008251173248928193754, −9.175546382853204689716082168852, −8.656867227583233678012818174756, −8.082212802889382388188676137660, −7.74009858595133668943213493996, −7.29070272123673306221124555605, −6.64286022823718963402047851135, −6.09558515130673864815367142993, −5.54934240217595040540496676447, −4.90444653422574791380984398340, −4.37043664329120792258997883679, −3.44887876808808918031065623865, −2.57822411195881754594425958622, −2.26985252840327432910532308213, −0.76476526582840185520577322367,
0.76476526582840185520577322367, 2.26985252840327432910532308213, 2.57822411195881754594425958622, 3.44887876808808918031065623865, 4.37043664329120792258997883679, 4.90444653422574791380984398340, 5.54934240217595040540496676447, 6.09558515130673864815367142993, 6.64286022823718963402047851135, 7.29070272123673306221124555605, 7.74009858595133668943213493996, 8.082212802889382388188676137660, 8.656867227583233678012818174756, 9.175546382853204689716082168852, 9.458409839008251173248928193754
