| L(s) = 1 | − 1.86·2-s − 0.722·3-s + 1.46·4-s − 1.11·5-s + 1.34·6-s − 0.401·7-s + 0.993·8-s − 2.47·9-s + 2.07·10-s + 0.789·11-s − 1.05·12-s + 0.855·13-s + 0.746·14-s + 0.804·15-s − 4.78·16-s + 3.46·17-s + 4.61·18-s + 1.74·19-s − 1.63·20-s + 0.289·21-s − 1.46·22-s − 1.52·23-s − 0.718·24-s − 3.76·25-s − 1.59·26-s + 3.95·27-s − 0.588·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s − 0.417·3-s + 0.733·4-s − 0.497·5-s + 0.549·6-s − 0.151·7-s + 0.351·8-s − 0.825·9-s + 0.655·10-s + 0.237·11-s − 0.305·12-s + 0.237·13-s + 0.199·14-s + 0.207·15-s − 1.19·16-s + 0.839·17-s + 1.08·18-s + 0.400·19-s − 0.364·20-s + 0.0632·21-s − 0.313·22-s − 0.318·23-s − 0.146·24-s − 0.752·25-s − 0.312·26-s + 0.761·27-s − 0.111·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4183433676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4183433676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 3 | \( 1 + 0.722T + 3T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 7 | \( 1 + 0.401T + 7T^{2} \) |
| 11 | \( 1 - 0.789T + 11T^{2} \) |
| 13 | \( 1 - 0.855T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 + 3.97T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 + 5.42T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 + 1.68T + 53T^{2} \) |
| 59 | \( 1 + 8.31T + 59T^{2} \) |
| 61 | \( 1 - 8.40T + 61T^{2} \) |
| 67 | \( 1 - 7.29T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 6.33T + 83T^{2} \) |
| 89 | \( 1 + 9.45T + 89T^{2} \) |
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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
−7.69121843237396330647196618664, −7.40402179951473612607530676315, −6.44754281811338864749809130466, −5.77502610744387206608884239337, −5.10660387560978031569320215787, −4.09367817526290541571254884504, −3.42080204372329856381201751667, −2.37094304887818315643601897494, −1.36236851237847946207232385935, −0.41887171739714398627787364962,
0.41887171739714398627787364962, 1.36236851237847946207232385935, 2.37094304887818315643601897494, 3.42080204372329856381201751667, 4.09367817526290541571254884504, 5.10660387560978031569320215787, 5.77502610744387206608884239337, 6.44754281811338864749809130466, 7.40402179951473612607530676315, 7.69121843237396330647196618664
