| L(s) = 1 | + 1.67·2-s − 5.19·4-s + 7·7-s − 22.1·8-s + 57.5·11-s − 45.5·13-s + 11.7·14-s + 4.52·16-s − 92.0·17-s + 125.·19-s + 96.4·22-s − 158.·23-s − 76.2·26-s − 36.3·28-s + 40.1·29-s + 49.5·31-s + 184.·32-s − 154.·34-s + 231.·37-s + 209.·38-s − 169.·41-s − 147.·43-s − 299.·44-s − 265.·46-s + 67.0·47-s + 49·49-s + 236.·52-s + ⋯ |
| L(s) = 1 | + 0.592·2-s − 0.649·4-s + 0.377·7-s − 0.976·8-s + 1.57·11-s − 0.971·13-s + 0.223·14-s + 0.0707·16-s − 1.31·17-s + 1.51·19-s + 0.934·22-s − 1.43·23-s − 0.575·26-s − 0.245·28-s + 0.257·29-s + 0.287·31-s + 1.01·32-s − 0.777·34-s + 1.02·37-s + 0.895·38-s − 0.645·41-s − 0.522·43-s − 1.02·44-s − 0.851·46-s + 0.208·47-s + 0.142·49-s + 0.630·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 1.67T + 8T^{2} \) |
| 11 | \( 1 - 57.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 92.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 40.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 49.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 231.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 169.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 268.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 240.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 90.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 406.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 330.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 546.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 25.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 376.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 942.T + 9.12e5T^{2} \) |
| 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
−8.745902683300732360405317097746, −7.923670885801050249561884873877, −6.89499438734062235394478706842, −6.13838204521935852035002906342, −5.18965847949627714622762268655, −4.41089372216940190857845530319, −3.80028862146580168857609324720, −2.63250899334183724154450632896, −1.32989529005646332941396228847, 0,
1.32989529005646332941396228847, 2.63250899334183724154450632896, 3.80028862146580168857609324720, 4.41089372216940190857845530319, 5.18965847949627714622762268655, 6.13838204521935852035002906342, 6.89499438734062235394478706842, 7.923670885801050249561884873877, 8.745902683300732360405317097746
