| L(s) = 1 | + 5.24·2-s + 19.4·4-s − 5·5-s − 18.4·7-s + 60.0·8-s − 26.2·10-s − 11.5·11-s − 34.4·13-s − 96.6·14-s + 159.·16-s − 70.7·17-s + 3.52·19-s − 97.3·20-s − 60.4·22-s + 18.3·23-s + 25·25-s − 180.·26-s − 359.·28-s − 29·29-s − 120.·31-s + 353.·32-s − 370.·34-s + 92.2·35-s − 182.·37-s + 18.4·38-s − 300.·40-s − 74.4·41-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.43·4-s − 0.447·5-s − 0.996·7-s + 2.65·8-s − 0.828·10-s − 0.316·11-s − 0.735·13-s − 1.84·14-s + 2.48·16-s − 1.00·17-s + 0.0425·19-s − 1.08·20-s − 0.585·22-s + 0.166·23-s + 0.200·25-s − 1.36·26-s − 2.42·28-s − 0.185·29-s − 0.698·31-s + 1.95·32-s − 1.87·34-s + 0.445·35-s − 0.810·37-s + 0.0787·38-s − 1.18·40-s − 0.283·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 5.24T + 8T^{2} \) |
| 7 | \( 1 + 18.4T + 343T^{2} \) |
| 11 | \( 1 + 11.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 70.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 3.52T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18.3T + 1.21e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 74.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 405.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 327.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 120.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 34.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 671.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 873.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 432.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 548.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 511.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 788.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 967.T + 9.12e5T^{2} \) |
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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
−8.834295645648886903191391959806, −7.58922317685896206472814650065, −6.91098921579089286727680459426, −6.28452774099409358165120992919, −5.29948942810818715251206358019, −4.59764256450685490673693803676, −3.65683635501457198798157905983, −2.97211939548308148220912384280, −1.98867930045874675762421816949, 0,
1.98867930045874675762421816949, 2.97211939548308148220912384280, 3.65683635501457198798157905983, 4.59764256450685490673693803676, 5.29948942810818715251206358019, 6.28452774099409358165120992919, 6.91098921579089286727680459426, 7.58922317685896206472814650065, 8.834295645648886903191391959806
