| L(s) = 1 | − 8.62·3-s − 15.0·5-s + 7·7-s + 47.4·9-s + 3.73·11-s + 52.1·13-s + 130.·15-s + 6.47·17-s + 99.4·19-s − 60.3·21-s − 153.·23-s + 102.·25-s − 175.·27-s − 292.·29-s − 311.·31-s − 32.2·33-s − 105.·35-s − 270.·37-s − 449.·39-s − 41·41-s + 332.·43-s − 715.·45-s + 204.·47-s + 49·49-s − 55.8·51-s + 201.·53-s − 56.4·55-s + ⋯ |
| L(s) = 1 | − 1.65·3-s − 1.34·5-s + 0.377·7-s + 1.75·9-s + 0.102·11-s + 1.11·13-s + 2.24·15-s + 0.0923·17-s + 1.20·19-s − 0.627·21-s − 1.38·23-s + 0.822·25-s − 1.25·27-s − 1.87·29-s − 1.80·31-s − 0.170·33-s − 0.510·35-s − 1.20·37-s − 1.84·39-s − 0.156·41-s + 1.17·43-s − 2.37·45-s + 0.634·47-s + 0.142·49-s − 0.153·51-s + 0.523·53-s − 0.138·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5563443687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5563443687\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 + 8.62T + 27T^{2} \) |
| 5 | \( 1 + 15.0T + 125T^{2} \) |
| 11 | \( 1 - 3.73T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.47T + 4.91e3T^{2} \) |
| 19 | \( 1 - 99.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 292.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 311.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 270.T + 5.06e4T^{2} \) |
| 43 | \( 1 - 332.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 201.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 265.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 869.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 761.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 457.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 141.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 886.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 901.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.31e3T + 9.12e5T^{2} \) |
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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
−9.545799626930425301892308505804, −8.478997671966747972929121672533, −7.51113221645341326652342874193, −7.04711907496603925353011666468, −5.75525261877814697123013576459, −5.43301455230421793184208986372, −4.11807120315917223591222683522, −3.69159160774967831437863574106, −1.58305400261854358529726797853, −0.43953993823576415079508070258,
0.43953993823576415079508070258, 1.58305400261854358529726797853, 3.69159160774967831437863574106, 4.11807120315917223591222683522, 5.43301455230421793184208986372, 5.75525261877814697123013576459, 7.04711907496603925353011666468, 7.51113221645341326652342874193, 8.478997671966747972929121672533, 9.545799626930425301892308505804
