| L(s) = 1 | − 39.8·2-s + 138.·3-s + 1.07e3·4-s + 553.·5-s − 5.50e3·6-s + 2.40e3·7-s − 2.23e4·8-s − 585.·9-s − 2.20e4·10-s − 1.26e4·11-s + 1.48e5·12-s − 2.85e4·13-s − 9.55e4·14-s + 7.65e4·15-s + 3.39e5·16-s + 4.11e5·17-s + 2.33e4·18-s + 6.60e5·19-s + 5.94e5·20-s + 3.31e5·21-s + 5.02e5·22-s + 2.32e6·23-s − 3.08e6·24-s − 1.64e6·25-s + 1.13e6·26-s − 2.80e6·27-s + 2.57e6·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.985·3-s + 2.09·4-s + 0.396·5-s − 1.73·6-s + 0.377·7-s − 1.92·8-s − 0.0297·9-s − 0.696·10-s − 0.259·11-s + 2.06·12-s − 0.277·13-s − 0.665·14-s + 0.390·15-s + 1.29·16-s + 1.19·17-s + 0.0523·18-s + 1.16·19-s + 0.830·20-s + 0.372·21-s + 0.457·22-s + 1.73·23-s − 1.89·24-s − 0.843·25-s + 0.487·26-s − 1.01·27-s + 0.792·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.428187067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.428187067\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 + 39.8T + 512T^{2} \) |
| 3 | \( 1 - 138.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 553.T + 1.95e6T^{2} \) |
| 11 | \( 1 + 1.26e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 4.11e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.60e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.32e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.99e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.54e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.46e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.21e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.23e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.94e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.46e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.19e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.74e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.94e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.88e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 7.11e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.60e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.08e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.99e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.73e8T + 7.60e17T^{2} \) |
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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
−11.84038808284935966169831415201, −10.79722581685827591918060925223, −9.613451738941684853003415195617, −9.111426202666982878547709778879, −7.918435831817808057426946208500, −7.36010612362420473645104884921, −5.58605904492378478061106062925, −3.16436343052513601944475829754, −2.05686895660745202905172783682, −0.860316376261049036105543361877,
0.860316376261049036105543361877, 2.05686895660745202905172783682, 3.16436343052513601944475829754, 5.58605904492378478061106062925, 7.36010612362420473645104884921, 7.918435831817808057426946208500, 9.111426202666982878547709778879, 9.613451738941684853003415195617, 10.79722581685827591918060925223, 11.84038808284935966169831415201
