| L(s) = 1 | + 3-s + 5.12·7-s + 9-s − 2·11-s − 5.12·13-s + 1.12·17-s − 5.12·19-s + 5.12·21-s − 5.12·23-s + 27-s − 8.24·29-s + 7.12·31-s − 2·33-s + 5.12·37-s − 5.12·39-s − 2·41-s − 6.24·43-s − 13.1·47-s + 19.2·49-s + 1.12·51-s − 10·53-s − 5.12·57-s − 6·59-s − 2·61-s + 5.12·63-s − 6.24·67-s − 5.12·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.93·7-s + 0.333·9-s − 0.603·11-s − 1.42·13-s + 0.272·17-s − 1.17·19-s + 1.11·21-s − 1.06·23-s + 0.192·27-s − 1.53·29-s + 1.27·31-s − 0.348·33-s + 0.842·37-s − 0.820·39-s − 0.312·41-s − 0.952·43-s − 1.91·47-s + 2.74·49-s + 0.157·51-s − 1.37·53-s − 0.678·57-s − 0.781·59-s − 0.256·61-s + 0.645·63-s − 0.763·67-s − 0.616·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.68000445049182951241916863412, −6.82216966543961239273177646666, −5.91044808789046285880545403631, −4.98285669183716440274428081484, −4.71794775404622438084136123861, −3.96122367147727515843844431394, −2.83438031636290181879678904414, −2.08206522742949308385468259978, −1.57809938123855289726341672185, 0,
1.57809938123855289726341672185, 2.08206522742949308385468259978, 2.83438031636290181879678904414, 3.96122367147727515843844431394, 4.71794775404622438084136123861, 4.98285669183716440274428081484, 5.91044808789046285880545403631, 6.82216966543961239273177646666, 7.68000445049182951241916863412
