| L(s) = 1 | + 654.·2-s − 6.56e3·3-s + 2.97e5·4-s + 3.90e5·5-s − 4.29e6·6-s + 3.59e5·7-s + 1.08e8·8-s + 4.30e7·9-s + 2.55e8·10-s + 1.12e9·11-s − 1.95e9·12-s − 3.56e9·13-s + 2.35e8·14-s − 2.56e9·15-s + 3.22e10·16-s + 4.86e10·17-s + 2.81e10·18-s + 5.83e10·19-s + 1.16e11·20-s − 2.35e9·21-s + 7.36e11·22-s − 1.76e11·23-s − 7.14e11·24-s + 1.52e11·25-s − 2.33e12·26-s − 2.82e11·27-s + 1.06e11·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 0.577·3-s + 2.26·4-s + 0.447·5-s − 1.04·6-s + 0.0235·7-s + 2.29·8-s + 0.333·9-s + 0.808·10-s + 1.58·11-s − 1.30·12-s − 1.21·13-s + 0.0426·14-s − 0.258·15-s + 1.87·16-s + 1.68·17-s + 0.602·18-s + 0.787·19-s + 1.01·20-s − 0.0136·21-s + 2.86·22-s − 0.470·23-s − 1.32·24-s + 0.200·25-s − 2.19·26-s − 0.192·27-s + 0.0535·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(5.825549425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.825549425\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 5 | \( 1 - 3.90e5T \) |
| good | 2 | \( 1 - 654.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 3.59e5T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.12e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.56e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 4.86e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 5.83e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 1.76e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 2.78e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 6.55e9T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.71e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 1.87e12T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.01e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.15e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 8.28e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.11e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.77e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 5.09e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.44e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 3.88e14T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.71e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 7.42e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 1.06e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 4.95e15T + 5.95e33T^{2} \) |
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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
−14.67172192003275107833994626270, −13.97161359538279429050401978530, −12.32386849793630363069249667906, −11.78808766978672665751866782466, −9.983255526068871510643920138242, −7.10173802901433132419318993940, −5.89493980222124206734806630661, −4.75979629477430724444112079692, −3.28192418132394583377725019418, −1.49889297665907234001738396594,
1.49889297665907234001738396594, 3.28192418132394583377725019418, 4.75979629477430724444112079692, 5.89493980222124206734806630661, 7.10173802901433132419318993940, 9.983255526068871510643920138242, 11.78808766978672665751866782466, 12.32386849793630363069249667906, 13.97161359538279429050401978530, 14.67172192003275107833994626270
