| L(s) = 1 | + 0.0457·2-s − 10.2·3-s − 7.99·4-s + 5·5-s − 0.467·6-s − 33.8·7-s − 0.732·8-s + 77.4·9-s + 0.228·10-s − 21.0·11-s + 81.7·12-s + 32.8·13-s − 1.54·14-s − 51.0·15-s + 63.9·16-s − 28.7·17-s + 3.54·18-s + 67.2·19-s − 39.9·20-s + 346.·21-s − 0.962·22-s + 23·23-s + 7.48·24-s + 25·25-s + 1.50·26-s − 515.·27-s + 270.·28-s + ⋯ |
| L(s) = 1 | + 0.0161·2-s − 1.96·3-s − 0.999·4-s + 0.447·5-s − 0.0318·6-s − 1.82·7-s − 0.0323·8-s + 2.86·9-s + 0.00723·10-s − 0.576·11-s + 1.96·12-s + 0.701·13-s − 0.0295·14-s − 0.879·15-s + 0.999·16-s − 0.409·17-s + 0.0464·18-s + 0.812·19-s − 0.447·20-s + 3.59·21-s − 0.00932·22-s + 0.208·23-s + 0.0636·24-s + 0.200·25-s + 0.0113·26-s − 3.67·27-s + 1.82·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4096378918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4096378918\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 0.0457T + 8T^{2} \) |
| 3 | \( 1 + 10.2T + 27T^{2} \) |
| 7 | \( 1 + 33.8T + 343T^{2} \) |
| 11 | \( 1 + 21.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 28.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 199.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 2.47T + 2.97e4T^{2} \) |
| 37 | \( 1 + 173.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 198.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 33.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 556.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 26.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 190.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 745.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 742.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 715.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 766.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 683.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−13.01900088309646621380526718782, −12.25539238915022634107517798230, −10.88165945493859993582384400909, −10.01058547559569213935077954047, −9.279927742719726671081680563055, −7.13545668566898270368315203117, −6.02491637508595939800395644725, −5.35056184412338496762954718137, −3.84710920530781106597727684870, −0.60087811501644249758278501196,
0.60087811501644249758278501196, 3.84710920530781106597727684870, 5.35056184412338496762954718137, 6.02491637508595939800395644725, 7.13545668566898270368315203117, 9.279927742719726671081680563055, 10.01058547559569213935077954047, 10.88165945493859993582384400909, 12.25539238915022634107517798230, 13.01900088309646621380526718782
