| L(s) = 1 | + 2.48·3-s − 3.41·5-s − 3.75·7-s + 3.16·9-s − 6.39·11-s + 1.29·13-s − 8.47·15-s − 1.70·17-s − 7.55·19-s − 9.32·21-s − 7.46·23-s + 6.65·25-s + 0.397·27-s + 10.4·29-s + 8.25·31-s − 15.8·33-s + 12.8·35-s + 2.33·37-s + 3.21·39-s − 4.86·41-s + 2.43·43-s − 10.7·45-s + 4.90·47-s + 7.11·49-s − 4.24·51-s + 3.51·53-s + 21.8·55-s + ⋯ |
| L(s) = 1 | + 1.43·3-s − 1.52·5-s − 1.42·7-s + 1.05·9-s − 1.92·11-s + 0.359·13-s − 2.18·15-s − 0.414·17-s − 1.73·19-s − 2.03·21-s − 1.55·23-s + 1.33·25-s + 0.0765·27-s + 1.93·29-s + 1.48·31-s − 2.76·33-s + 2.16·35-s + 0.384·37-s + 0.515·39-s − 0.759·41-s + 0.371·43-s − 1.60·45-s + 0.715·47-s + 1.01·49-s − 0.593·51-s + 0.483·53-s + 2.94·55-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
| \(L(1)\) |
\(\approx\) |
\(0.7176673508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7176673508\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 + 3.75T + 7T^{2} \) |
| 11 | \( 1 + 6.39T + 11T^{2} \) |
| 13 | \( 1 - 1.29T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 7.55T + 19T^{2} \) |
| 23 | \( 1 + 7.46T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 8.25T + 31T^{2} \) |
| 37 | \( 1 - 2.33T + 37T^{2} \) |
| 41 | \( 1 + 4.86T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 - 3.51T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 8.62T + 61T^{2} \) |
| 67 | \( 1 + 5.49T + 67T^{2} \) |
| 71 | \( 1 + 9.14T + 71T^{2} \) |
| 73 | \( 1 + 7.32T + 73T^{2} \) |
| 79 | \( 1 - 2.73T + 79T^{2} \) |
| 83 | \( 1 - 8.20T + 83T^{2} \) |
| 89 | \( 1 + 0.0164T + 89T^{2} \) |
| 97 | \( 1 + 4.62T + 97T^{2} \) |
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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
−7.932460442482258476871852750689, −7.47017412857846037298969816816, −6.54305195888376111631638350017, −5.99001729888786741935127984743, −4.49832232152062051998761774333, −4.27466528217846445195924978957, −3.25760146976918517278673224972, −2.89444937968647919152849481525, −2.21404008615029448398926110765, −0.35574057805797513156309114921,
0.35574057805797513156309114921, 2.21404008615029448398926110765, 2.89444937968647919152849481525, 3.25760146976918517278673224972, 4.27466528217846445195924978957, 4.49832232152062051998761774333, 5.99001729888786741935127984743, 6.54305195888376111631638350017, 7.47017412857846037298969816816, 7.932460442482258476871852750689
