| L(s) = 1 | + 3.04·5-s + 0.418·7-s − 3·9-s + 5.27·11-s + 2.62·13-s + 3.27·17-s − 19-s + 3.46·23-s + 4.27·25-s + 2.62·29-s − 6.09·31-s + 1.27·35-s + 9.55·37-s − 4.54·41-s + 2.72·43-s − 9.13·45-s + 0.418·47-s − 6.82·49-s + 10.3·53-s + 16.0·55-s − 6.54·59-s − 3.04·61-s − 1.25·63-s + 8·65-s + 6.54·67-s − 11.3·71-s + 3.27·73-s + ⋯ |
| L(s) = 1 | + 1.36·5-s + 0.158·7-s − 9-s + 1.59·11-s + 0.728·13-s + 0.794·17-s − 0.229·19-s + 0.722·23-s + 0.854·25-s + 0.487·29-s − 1.09·31-s + 0.215·35-s + 1.57·37-s − 0.710·41-s + 0.415·43-s − 1.36·45-s + 0.0610·47-s − 0.974·49-s + 1.42·53-s + 2.16·55-s − 0.852·59-s − 0.389·61-s − 0.158·63-s + 0.992·65-s + 0.800·67-s − 1.34·71-s + 0.383·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.995068083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.995068083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 3.04T + 5T^{2} \) |
| 7 | \( 1 - 0.418T + 7T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 - 9.55T + 37T^{2} \) |
| 41 | \( 1 + 4.54T + 41T^{2} \) |
| 43 | \( 1 - 2.72T + 43T^{2} \) |
| 47 | \( 1 - 0.418T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6.54T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 - 6.54T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 3.27T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.578582897701666205209392999357, −7.49537788852050331480628096356, −6.60518733897276829733918562141, −6.00193545571285625840440561934, −5.63181212292909462646689275521, −4.63719254910866709713489906233, −3.64587161388860671380138964950, −2.84673044759999219384429092955, −1.80592485118276271436380260710, −1.03704251960549542422704917527,
1.03704251960549542422704917527, 1.80592485118276271436380260710, 2.84673044759999219384429092955, 3.64587161388860671380138964950, 4.63719254910866709713489906233, 5.63181212292909462646689275521, 6.00193545571285625840440561934, 6.60518733897276829733918562141, 7.49537788852050331480628096356, 8.578582897701666205209392999357
