| L(s) = 1 | − 1.59·3-s + 2.75·5-s − 1.46·7-s − 0.462·9-s + 5.86·11-s − 6.69·13-s − 4.38·15-s − 6.07·17-s + 1.48·19-s + 2.33·21-s − 4.40·23-s + 2.58·25-s + 5.51·27-s + 8.17·29-s + 8.84·31-s − 9.35·33-s − 4.03·35-s − 7.89·37-s + 10.6·39-s + 5.35·41-s + 8.36·43-s − 1.27·45-s − 9.07·47-s − 4.85·49-s + 9.67·51-s + 12.3·53-s + 16.1·55-s + ⋯ |
| L(s) = 1 | − 0.919·3-s + 1.23·5-s − 0.553·7-s − 0.154·9-s + 1.76·11-s − 1.85·13-s − 1.13·15-s − 1.47·17-s + 0.340·19-s + 0.508·21-s − 0.919·23-s + 0.517·25-s + 1.06·27-s + 1.51·29-s + 1.58·31-s − 1.62·33-s − 0.681·35-s − 1.29·37-s + 1.70·39-s + 0.836·41-s + 1.27·43-s − 0.189·45-s − 1.32·47-s − 0.693·49-s + 1.35·51-s + 1.69·53-s + 2.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 1.59T + 3T^{2} \) |
| 5 | \( 1 - 2.75T + 5T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 - 5.86T + 11T^{2} \) |
| 13 | \( 1 + 6.69T + 13T^{2} \) |
| 17 | \( 1 + 6.07T + 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 + 4.40T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 - 8.84T + 31T^{2} \) |
| 37 | \( 1 + 7.89T + 37T^{2} \) |
| 41 | \( 1 - 5.35T + 41T^{2} \) |
| 43 | \( 1 - 8.36T + 43T^{2} \) |
| 47 | \( 1 + 9.07T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 2.57T + 59T^{2} \) |
| 61 | \( 1 - 3.03T + 61T^{2} \) |
| 67 | \( 1 + 1.68T + 67T^{2} \) |
| 71 | \( 1 + 4.21T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + 8.75T + 89T^{2} \) |
| 97 | \( 1 - 2.54T + 97T^{2} \) |
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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
−7.03928868416339725170757491273, −6.63715191077240545801847505109, −6.20303359083087446286255888979, −5.53220937919851126438080434505, −4.71896682310990434438816768733, −4.19145158346584009038112992317, −2.84713228312377016688145660781, −2.26294848691749785915379774451, −1.18369158068410716079231707868, 0,
1.18369158068410716079231707868, 2.26294848691749785915379774451, 2.84713228312377016688145660781, 4.19145158346584009038112992317, 4.71896682310990434438816768733, 5.53220937919851126438080434505, 6.20303359083087446286255888979, 6.63715191077240545801847505109, 7.03928868416339725170757491273
