| L(s) = 1 | − 3·3-s + 12.9·5-s − 1.76·7-s + 9·9-s − 25.1·11-s + 13·13-s − 38.8·15-s − 62.6·17-s − 139.·19-s + 5.29·21-s + 56.6·23-s + 42.9·25-s − 27·27-s + 75.4·29-s + 71.2·31-s + 75.5·33-s − 22.8·35-s + 55.7·37-s − 39·39-s − 40.0·41-s − 14.7·43-s + 116.·45-s − 531.·47-s − 339.·49-s + 187.·51-s + 368.·53-s − 326.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.15·5-s − 0.0952·7-s + 0.333·9-s − 0.690·11-s + 0.277·13-s − 0.669·15-s − 0.893·17-s − 1.68·19-s + 0.0550·21-s + 0.513·23-s + 0.343·25-s − 0.192·27-s + 0.482·29-s + 0.413·31-s + 0.398·33-s − 0.110·35-s + 0.247·37-s − 0.160·39-s − 0.152·41-s − 0.0524·43-s + 0.386·45-s − 1.64·47-s − 0.990·49-s + 0.515·51-s + 0.953·53-s − 0.800·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 12.9T + 125T^{2} \) |
| 7 | \( 1 + 1.76T + 343T^{2} \) |
| 11 | \( 1 + 25.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 62.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 56.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 75.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 71.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 55.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 40.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 14.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 531.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 368.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 165.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 145.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 901.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 345.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 292.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 722.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 565.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 275.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.82e3T + 9.12e5T^{2} \) |
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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
−9.936692305644825777680329004406, −8.991295766749677425617197667542, −8.103430200097005327058018624256, −6.71681765919140209181995452298, −6.22855775692313797677855757660, −5.24335654653058596631574989829, −4.33266071353213722958461530170, −2.70202224438397554846499407021, −1.64850874496815260019301476871, 0,
1.64850874496815260019301476871, 2.70202224438397554846499407021, 4.33266071353213722958461530170, 5.24335654653058596631574989829, 6.22855775692313797677855757660, 6.71681765919140209181995452298, 8.103430200097005327058018624256, 8.991295766749677425617197667542, 9.936692305644825777680329004406
