| L(s) = 1 | + 3.10·3-s + 2.27·5-s − 4.74·7-s + 6.61·9-s − 2.15·11-s − 3.51·13-s + 7.05·15-s − 2.38·17-s − 19-s − 14.7·21-s − 1.33·23-s + 0.180·25-s + 11.2·27-s − 8.75·29-s − 2.42·31-s − 6.67·33-s − 10.8·35-s − 4.49·37-s − 10.9·39-s − 8.53·41-s − 4.64·43-s + 15.0·45-s + 8.04·47-s + 15.5·49-s − 7.40·51-s − 3.03·53-s − 4.89·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s + 1.01·5-s − 1.79·7-s + 2.20·9-s − 0.648·11-s − 0.975·13-s + 1.82·15-s − 0.578·17-s − 0.229·19-s − 3.21·21-s − 0.278·23-s + 0.0360·25-s + 2.15·27-s − 1.62·29-s − 0.435·31-s − 1.16·33-s − 1.82·35-s − 0.739·37-s − 1.74·39-s − 1.33·41-s − 0.708·43-s + 2.24·45-s + 1.17·47-s + 2.22·49-s − 1.03·51-s − 0.416·53-s − 0.660·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 + 4.74T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 + 3.51T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 23 | \( 1 + 1.33T + 23T^{2} \) |
| 29 | \( 1 + 8.75T + 29T^{2} \) |
| 31 | \( 1 + 2.42T + 31T^{2} \) |
| 37 | \( 1 + 4.49T + 37T^{2} \) |
| 41 | \( 1 + 8.53T + 41T^{2} \) |
| 43 | \( 1 + 4.64T + 43T^{2} \) |
| 47 | \( 1 - 8.04T + 47T^{2} \) |
| 53 | \( 1 + 3.03T + 53T^{2} \) |
| 59 | \( 1 - 3.67T + 59T^{2} \) |
| 61 | \( 1 + 4.35T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 9.07T + 71T^{2} \) |
| 73 | \( 1 - 8.69T + 73T^{2} \) |
| 83 | \( 1 - 5.57T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 - 9.29T + 97T^{2} \) |
| 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
−7.60053487178653095486696581777, −7.22833939127359505206277334541, −6.43282733660114278325715997532, −5.69671119156519313333783624360, −4.69951786273657550398371301599, −3.62713963555610836028684384900, −3.19782250197667604060273372264, −2.30143253853859699318653275791, −1.94085131870609323282240098310, 0,
1.94085131870609323282240098310, 2.30143253853859699318653275791, 3.19782250197667604060273372264, 3.62713963555610836028684384900, 4.69951786273657550398371301599, 5.69671119156519313333783624360, 6.43282733660114278325715997532, 7.22833939127359505206277334541, 7.60053487178653095486696581777
