| L(s) = 1 | + 6.41·3-s − 20.6·5-s − 13.3·7-s + 14.1·9-s + 11.4·11-s + 48.2·13-s − 132.·15-s − 33.5·19-s − 85.8·21-s + 17.7·23-s + 300.·25-s − 82.2·27-s + 204.·29-s + 177.·31-s + 73.7·33-s + 275.·35-s − 26.8·37-s + 309.·39-s + 487.·41-s − 352.·43-s − 292.·45-s − 299.·47-s − 163.·49-s + 347.·53-s − 236.·55-s − 215.·57-s − 511.·59-s + ⋯ |
| L(s) = 1 | + 1.23·3-s − 1.84·5-s − 0.722·7-s + 0.525·9-s + 0.314·11-s + 1.02·13-s − 2.27·15-s − 0.404·19-s − 0.892·21-s + 0.161·23-s + 2.40·25-s − 0.586·27-s + 1.31·29-s + 1.02·31-s + 0.388·33-s + 1.33·35-s − 0.119·37-s + 1.27·39-s + 1.85·41-s − 1.24·43-s − 0.968·45-s − 0.929·47-s − 0.478·49-s + 0.899·53-s − 0.580·55-s − 0.499·57-s − 1.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 6.41T + 27T^{2} \) |
| 5 | \( 1 + 20.6T + 125T^{2} \) |
| 7 | \( 1 + 13.3T + 343T^{2} \) |
| 11 | \( 1 - 11.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.2T + 2.19e3T^{2} \) |
| 19 | \( 1 + 33.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 17.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 26.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 487.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 352.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 299.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 347.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 511.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 351.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 811.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 152.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 923.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 832.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 428.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 527.T + 9.12e5T^{2} \) |
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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
−8.274428041411636737510958915305, −7.78863729230776367280978169030, −6.87250831952599932960122003994, −6.19402621653525013861468324912, −4.65825303642452623463737593340, −3.97405375398697570039434501925, −3.30776488839785344310455020938, −2.75088318836399691642695519680, −1.16752742437239109158732187478, 0,
1.16752742437239109158732187478, 2.75088318836399691642695519680, 3.30776488839785344310455020938, 3.97405375398697570039434501925, 4.65825303642452623463737593340, 6.19402621653525013861468324912, 6.87250831952599932960122003994, 7.78863729230776367280978169030, 8.274428041411636737510958915305
