| L(s) = 1 | + 0.245·2-s − 7.93·4-s + 10.5·7-s − 3.91·8-s − 11·11-s − 63.0·13-s + 2.58·14-s + 62.5·16-s + 133.·17-s + 76.1·19-s − 2.70·22-s − 169.·23-s − 15.4·26-s − 83.3·28-s − 202.·29-s + 191.·31-s + 46.7·32-s + 32.9·34-s + 21.5·37-s + 18.7·38-s − 305.·41-s + 285.·43-s + 87.3·44-s − 41.7·46-s − 123.·47-s − 232.·49-s + 500.·52-s + ⋯ |
| L(s) = 1 | + 0.0869·2-s − 0.992·4-s + 0.567·7-s − 0.173·8-s − 0.301·11-s − 1.34·13-s + 0.0492·14-s + 0.977·16-s + 1.91·17-s + 0.919·19-s − 0.0262·22-s − 1.53·23-s − 0.116·26-s − 0.562·28-s − 1.29·29-s + 1.11·31-s + 0.258·32-s + 0.166·34-s + 0.0959·37-s + 0.0799·38-s − 1.16·41-s + 1.01·43-s + 0.299·44-s − 0.133·46-s − 0.384·47-s − 0.678·49-s + 1.33·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.454582222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.454582222\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 0.245T + 8T^{2} \) |
| 7 | \( 1 - 10.5T + 343T^{2} \) |
| 13 | \( 1 + 63.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 133.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 21.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 305.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 123.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 480.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 364.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 9.11T + 2.26e5T^{2} \) |
| 67 | \( 1 + 568.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 157.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 212.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 792.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 587.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 698.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.83e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.462585061199200945100749530910, −7.76817611819846356661867632284, −7.43225109707965461356240145780, −5.95849769433227488559561754232, −5.34288398877078349785668964196, −4.73677334854223854543104429276, −3.80051880879109879661376909897, −2.93212508217415879556031542837, −1.66673000011808765660518360403, −0.53951177719964469512361154629,
0.53951177719964469512361154629, 1.66673000011808765660518360403, 2.93212508217415879556031542837, 3.80051880879109879661376909897, 4.73677334854223854543104429276, 5.34288398877078349785668964196, 5.95849769433227488559561754232, 7.43225109707965461356240145780, 7.76817611819846356661867632284, 8.462585061199200945100749530910
