| L(s) = 1 | − 1.33·2-s + 2.94·3-s − 0.229·4-s + 3.80·5-s − 3.92·6-s + 7-s + 2.96·8-s + 5.68·9-s − 5.05·10-s − 2.75·11-s − 0.675·12-s − 3.31·13-s − 1.33·14-s + 11.2·15-s − 3.48·16-s + 1.64·17-s − 7.56·18-s − 7.09·19-s − 0.871·20-s + 2.94·21-s + 3.66·22-s − 2.81·23-s + 8.74·24-s + 9.45·25-s + 4.41·26-s + 7.90·27-s − 0.229·28-s + ⋯ |
| L(s) = 1 | − 0.940·2-s + 1.70·3-s − 0.114·4-s + 1.70·5-s − 1.60·6-s + 0.377·7-s + 1.04·8-s + 1.89·9-s − 1.59·10-s − 0.830·11-s − 0.195·12-s − 0.920·13-s − 0.355·14-s + 2.89·15-s − 0.872·16-s + 0.397·17-s − 1.78·18-s − 1.62·19-s − 0.194·20-s + 0.643·21-s + 0.780·22-s − 0.587·23-s + 1.78·24-s + 1.89·25-s + 0.865·26-s + 1.52·27-s − 0.0433·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.663714169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.663714169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 2 | \( 1 + 1.33T + 2T^{2} \) |
| 3 | \( 1 - 2.94T + 3T^{2} \) |
| 5 | \( 1 - 3.80T + 5T^{2} \) |
| 11 | \( 1 + 2.75T + 11T^{2} \) |
| 13 | \( 1 + 3.31T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 + 2.81T + 23T^{2} \) |
| 29 | \( 1 + 7.34T + 29T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 - 4.49T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 + 2.29T + 43T^{2} \) |
| 47 | \( 1 - 9.59T + 47T^{2} \) |
| 59 | \( 1 + 5.57T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 9.00T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
| 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
−10.74733318749013524543858608268, −9.984799826635509744023829660719, −9.529322016751245387441953524472, −8.733630139360847556288599826353, −8.017259798010844792875882348941, −7.13364892902578314029529805788, −5.55105423531623790695414869286, −4.27981033866118454147912333816, −2.47940104797334309850761992130, −1.85711021156526398753164263254,
1.85711021156526398753164263254, 2.47940104797334309850761992130, 4.27981033866118454147912333816, 5.55105423531623790695414869286, 7.13364892902578314029529805788, 8.017259798010844792875882348941, 8.733630139360847556288599826353, 9.529322016751245387441953524472, 9.984799826635509744023829660719, 10.74733318749013524543858608268
