| L(s) = 1 | − 8.71·3-s + 8.71·7-s + 49.0·9-s + 20·11-s − 52.3·13-s + 69.7·17-s + 84·19-s − 76.0·21-s + 61.0·23-s − 191.·27-s + 6·29-s + 224·31-s − 174.·33-s − 122.·37-s + 456.·39-s + 266·41-s + 305.·43-s − 374.·47-s − 267·49-s − 608.·51-s − 366.·53-s − 732.·57-s + 28·59-s − 182·61-s + 427.·63-s + 427.·67-s − 532.·69-s + ⋯ |
| L(s) = 1 | − 1.67·3-s + 0.470·7-s + 1.81·9-s + 0.548·11-s − 1.11·13-s + 0.995·17-s + 1.01·19-s − 0.789·21-s + 0.553·23-s − 1.36·27-s + 0.0384·29-s + 1.29·31-s − 0.919·33-s − 0.542·37-s + 1.87·39-s + 1.01·41-s + 1.08·43-s − 1.16·47-s − 0.778·49-s − 1.66·51-s − 0.948·53-s − 1.70·57-s + 0.0617·59-s − 0.382·61-s + 0.854·63-s + 0.778·67-s − 0.928·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.242218331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242218331\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 8.71T + 27T^{2} \) |
| 7 | \( 1 - 8.71T + 343T^{2} \) |
| 11 | \( 1 - 20T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 69.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84T + 6.85e3T^{2} \) |
| 23 | \( 1 - 61.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 224T + 2.97e4T^{2} \) |
| 37 | \( 1 + 122.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 266T + 6.89e4T^{2} \) |
| 43 | \( 1 - 305.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 374.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 366.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 28T + 2.05e5T^{2} \) |
| 61 | \( 1 + 182T + 2.26e5T^{2} \) |
| 67 | \( 1 - 427.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 408T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.08e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 48T + 4.93e5T^{2} \) |
| 83 | \( 1 - 200.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 557.T + 9.12e5T^{2} \) |
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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
−9.352167462564216184986929885345, −7.996762157185497206416355863187, −7.30182214397250391318753860008, −6.52604124749415489295408646395, −5.69196109395251164401986492028, −5.01578779829835891816491900882, −4.40011120645302949924776446216, −3.03290266387278896324212781959, −1.48417244285713430925029075148, −0.63964825998583942618250243213,
0.63964825998583942618250243213, 1.48417244285713430925029075148, 3.03290266387278896324212781959, 4.40011120645302949924776446216, 5.01578779829835891816491900882, 5.69196109395251164401986492028, 6.52604124749415489295408646395, 7.30182214397250391318753860008, 7.996762157185497206416355863187, 9.352167462564216184986929885345
