| L(s) = 1 | + 3.04·3-s − 5-s − 4.39·7-s + 6.29·9-s − 2.58·11-s − 1.49·13-s − 3.04·15-s + 4.70·17-s + 6.84·19-s − 13.3·21-s + 4.09·23-s + 25-s + 10.0·27-s + 8.88·29-s − 0.295·31-s − 7.89·33-s + 4.39·35-s + 37-s − 4.55·39-s − 1.27·41-s + 2.09·43-s − 6.29·45-s + 10.3·47-s + 12.2·49-s + 14.3·51-s − 2.68·53-s + 2.58·55-s + ⋯ |
| L(s) = 1 | + 1.75·3-s − 0.447·5-s − 1.65·7-s + 2.09·9-s − 0.780·11-s − 0.414·13-s − 0.786·15-s + 1.14·17-s + 1.57·19-s − 2.92·21-s + 0.854·23-s + 0.200·25-s + 1.93·27-s + 1.65·29-s − 0.0531·31-s − 1.37·33-s + 0.742·35-s + 0.164·37-s − 0.728·39-s − 0.199·41-s + 0.319·43-s − 0.937·45-s + 1.51·47-s + 1.75·49-s + 2.00·51-s − 0.368·53-s + 0.349·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.799241275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.799241275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 7 | \( 1 + 4.39T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 - 4.70T + 17T^{2} \) |
| 19 | \( 1 - 6.84T + 19T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 29 | \( 1 - 8.88T + 29T^{2} \) |
| 31 | \( 1 + 0.295T + 31T^{2} \) |
| 41 | \( 1 + 1.27T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 + 4.49T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 4.84T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 - 0.202T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 5.00T + 97T^{2} \) |
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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
−8.885975249439349583503087590250, −7.892360590630344494692200669452, −7.45958500755326962913747752262, −6.83015880927161937814741122323, −5.69586619768726614417620536628, −4.64200871236523026469160444665, −3.54948838361573722234381983242, −3.07242452969188332686073943132, −2.60458585228171349260420342968, −0.954445954547667238766250857007,
0.954445954547667238766250857007, 2.60458585228171349260420342968, 3.07242452969188332686073943132, 3.54948838361573722234381983242, 4.64200871236523026469160444665, 5.69586619768726614417620536628, 6.83015880927161937814741122323, 7.45958500755326962913747752262, 7.892360590630344494692200669452, 8.885975249439349583503087590250
