| L(s) = 1 | + 2.69·2-s + 5.24·4-s + 3.09·5-s + 7-s + 8.72·8-s + 8.34·10-s + 11-s − 13-s + 2.69·14-s + 12.9·16-s + 0.282·17-s − 6.34·19-s + 16.2·20-s + 2.69·22-s + 3.52·23-s + 4.60·25-s − 2.69·26-s + 5.24·28-s − 5.03·29-s + 7.19·31-s + 17.5·32-s + 0.759·34-s + 3.09·35-s − 11.6·37-s − 17.0·38-s + 27.0·40-s + 1.69·41-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.62·4-s + 1.38·5-s + 0.377·7-s + 3.08·8-s + 2.63·10-s + 0.301·11-s − 0.277·13-s + 0.719·14-s + 3.24·16-s + 0.0684·17-s − 1.45·19-s + 3.63·20-s + 0.573·22-s + 0.734·23-s + 0.921·25-s − 0.527·26-s + 0.990·28-s − 0.934·29-s + 1.29·31-s + 3.09·32-s + 0.130·34-s + 0.523·35-s − 1.91·37-s − 2.76·38-s + 4.27·40-s + 0.264·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.88685406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.88685406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 - 3.09T + 5T^{2} \) |
| 17 | \( 1 - 0.282T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 23 | \( 1 - 3.52T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 - 7.19T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 1.69T + 41T^{2} \) |
| 43 | \( 1 + 0.450T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 - 9.11T + 53T^{2} \) |
| 59 | \( 1 - 6.84T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 6.30T + 67T^{2} \) |
| 71 | \( 1 + 0.365T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 0.717T + 79T^{2} \) |
| 83 | \( 1 + 2.42T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 1.87T + 97T^{2} \) |
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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
−7.25903956851033649723501958895, −6.70372428644383750304069051039, −6.23447125903967002663044905888, −5.41287853752469931072203344907, −5.17379349251627198721454603554, −4.27799361917691625120391734641, −3.65893353020128905206947677647, −2.58072010880422205860392923673, −2.17296134403109244964220628004, −1.35450941415488019650534665961,
1.35450941415488019650534665961, 2.17296134403109244964220628004, 2.58072010880422205860392923673, 3.65893353020128905206947677647, 4.27799361917691625120391734641, 5.17379349251627198721454603554, 5.41287853752469931072203344907, 6.23447125903967002663044905888, 6.70372428644383750304069051039, 7.25903956851033649723501958895
