| L(s) = 1 | + 3-s − 17·5-s − 35·7-s − 26·9-s − 2·11-s − 13·13-s − 17·15-s − 19·17-s − 94·19-s − 35·21-s − 72·23-s + 164·25-s − 53·27-s − 246·29-s − 100·31-s − 2·33-s + 595·35-s + 11·37-s − 13·39-s − 280·41-s − 241·43-s + 442·45-s + 137·47-s + 882·49-s − 19·51-s + 232·53-s + 34·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 1.52·5-s − 1.88·7-s − 0.962·9-s − 0.0548·11-s − 0.277·13-s − 0.292·15-s − 0.271·17-s − 1.13·19-s − 0.363·21-s − 0.652·23-s + 1.31·25-s − 0.377·27-s − 1.57·29-s − 0.579·31-s − 0.0105·33-s + 2.87·35-s + 0.0488·37-s − 0.0533·39-s − 1.06·41-s − 0.854·43-s + 1.46·45-s + 0.425·47-s + 18/7·49-s − 0.0521·51-s + 0.601·53-s + 0.0833·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 + 17 T + p^{3} T^{2} \) |
| 7 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 19 T + p^{3} T^{2} \) |
| 19 | \( 1 + 94 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 246 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 11 T + p^{3} T^{2} \) |
| 41 | \( 1 + 280 T + p^{3} T^{2} \) |
| 43 | \( 1 + 241 T + p^{3} T^{2} \) |
| 47 | \( 1 - 137 T + p^{3} T^{2} \) |
| 53 | \( 1 - 232 T + p^{3} T^{2} \) |
| 59 | \( 1 - 386 T + p^{3} T^{2} \) |
| 61 | \( 1 + 64 T + p^{3} T^{2} \) |
| 67 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 71 | \( 1 - 55 T + p^{3} T^{2} \) |
| 73 | \( 1 + 838 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1016 T + p^{3} T^{2} \) |
| 83 | \( 1 + 420 T + p^{3} T^{2} \) |
| 89 | \( 1 + 934 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1154 T + p^{3} T^{2} \) |
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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
−8.893608821283458842455491846868, −8.201555032134678594757629789613, −7.22179551992714983732938468590, −6.50039080846642568797072053797, −5.49333246523924985896198621066, −3.98841798613362374558482410876, −3.52051600178003154029590761078, −2.48185677046734381778508634141, 0, 0,
2.48185677046734381778508634141, 3.52051600178003154029590761078, 3.98841798613362374558482410876, 5.49333246523924985896198621066, 6.50039080846642568797072053797, 7.22179551992714983732938468590, 8.201555032134678594757629789613, 8.893608821283458842455491846868
