| L(s) = 1 | − 5.93·2-s − 1.91·3-s + 3.20·4-s − 25·5-s + 11.3·6-s + 107.·7-s + 170.·8-s − 239.·9-s + 148.·10-s − 121·11-s − 6.12·12-s + 56.0·13-s − 638.·14-s + 47.8·15-s − 1.11e3·16-s − 393.·17-s + 1.41e3·18-s − 361·19-s − 80.0·20-s − 205.·21-s + 717.·22-s + 2.94e3·23-s − 327.·24-s + 625·25-s − 332.·26-s + 923.·27-s + 344.·28-s + ⋯ |
| L(s) = 1 | − 1.04·2-s − 0.122·3-s + 0.100·4-s − 0.447·5-s + 0.128·6-s + 0.829·7-s + 0.943·8-s − 0.984·9-s + 0.469·10-s − 0.301·11-s − 0.0122·12-s + 0.0919·13-s − 0.870·14-s + 0.0549·15-s − 1.09·16-s − 0.330·17-s + 1.03·18-s − 0.229·19-s − 0.0447·20-s − 0.101·21-s + 0.316·22-s + 1.16·23-s − 0.115·24-s + 0.200·25-s − 0.0963·26-s + 0.243·27-s + 0.0830·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 5.93T + 32T^{2} \) |
| 3 | \( 1 + 1.91T + 243T^{2} \) |
| 7 | \( 1 - 107.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 56.0T + 3.71e5T^{2} \) |
| 17 | \( 1 + 393.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.94e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.14e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.68e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.13e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.64e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.47e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.26e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.02e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.63e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.677381406033723874727009638291, −8.154443635886284939779284086600, −7.45468872749643587422171992254, −6.43672101375243256733978026165, −5.16736669621288525780328338994, −4.59155868745467246290207325330, −3.27747081818845266963322350740, −2.04605282829403178562740626372, −0.922893283530352727535815489273, 0,
0.922893283530352727535815489273, 2.04605282829403178562740626372, 3.27747081818845266963322350740, 4.59155868745467246290207325330, 5.16736669621288525780328338994, 6.43672101375243256733978026165, 7.45468872749643587422171992254, 8.154443635886284939779284086600, 8.677381406033723874727009638291
