| L(s) = 1 | + 2-s + 3.08·3-s + 4-s − 3.01·5-s + 3.08·6-s − 0.707·7-s + 8-s + 6.49·9-s − 3.01·10-s − 3.04·11-s + 3.08·12-s + 3.77·13-s − 0.707·14-s − 9.30·15-s + 16-s − 2.97·17-s + 6.49·18-s + 5.40·19-s − 3.01·20-s − 2.17·21-s − 3.04·22-s + 23-s + 3.08·24-s + 4.11·25-s + 3.77·26-s + 10.7·27-s − 0.707·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.77·3-s + 0.5·4-s − 1.35·5-s + 1.25·6-s − 0.267·7-s + 0.353·8-s + 2.16·9-s − 0.954·10-s − 0.918·11-s + 0.889·12-s + 1.04·13-s − 0.189·14-s − 2.40·15-s + 0.250·16-s − 0.722·17-s + 1.53·18-s + 1.24·19-s − 0.675·20-s − 0.475·21-s − 0.649·22-s + 0.208·23-s + 0.628·24-s + 0.823·25-s + 0.740·26-s + 2.07·27-s − 0.133·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.030011570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.030011570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 3.08T + 3T^{2} \) |
| 5 | \( 1 + 3.01T + 5T^{2} \) |
| 7 | \( 1 + 0.707T + 7T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 - 5.40T + 19T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 - 3.64T + 31T^{2} \) |
| 37 | \( 1 - 6.65T + 37T^{2} \) |
| 41 | \( 1 + 0.825T + 41T^{2} \) |
| 43 | \( 1 - 4.00T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 9.75T + 59T^{2} \) |
| 61 | \( 1 - 0.576T + 61T^{2} \) |
| 67 | \( 1 + 0.641T + 67T^{2} \) |
| 71 | \( 1 + 5.33T + 71T^{2} \) |
| 73 | \( 1 + 8.67T + 73T^{2} \) |
| 79 | \( 1 - 7.89T + 79T^{2} \) |
| 83 | \( 1 - 8.75T + 83T^{2} \) |
| 89 | \( 1 + 1.23T + 89T^{2} \) |
| 97 | \( 1 - 6.07T + 97T^{2} \) |
| 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
−7.983670326665016067610282867674, −7.57807488921545124611111957689, −6.91557936663704230432011571640, −5.99491208769809963111501206900, −4.78541241006469916999274047241, −4.24644465825175374480943775424, −3.48190708697921760467316814517, −3.05495831024361669594454099694, −2.31951743088683336657746287649, −1.00147828523648751667033509583,
1.00147828523648751667033509583, 2.31951743088683336657746287649, 3.05495831024361669594454099694, 3.48190708697921760467316814517, 4.24644465825175374480943775424, 4.78541241006469916999274047241, 5.99491208769809963111501206900, 6.91557936663704230432011571640, 7.57807488921545124611111957689, 7.983670326665016067610282867674
