| L(s) = 1 | + 2-s − 1.98·3-s + 4-s − 3.80·5-s − 1.98·6-s + 8-s + 0.947·9-s − 3.80·10-s + 5.77·11-s − 1.98·12-s − 1.39·13-s + 7.55·15-s + 16-s − 3.39·17-s + 0.947·18-s + 3.16·19-s − 3.80·20-s + 5.77·22-s − 8.54·23-s − 1.98·24-s + 9.45·25-s − 1.39·26-s + 4.07·27-s − 6.67·29-s + 7.55·30-s − 3.32·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.14·3-s + 0.5·4-s − 1.70·5-s − 0.811·6-s + 0.353·8-s + 0.315·9-s − 1.20·10-s + 1.73·11-s − 0.573·12-s − 0.385·13-s + 1.95·15-s + 0.250·16-s − 0.822·17-s + 0.223·18-s + 0.726·19-s − 0.850·20-s + 1.23·22-s − 1.78·23-s − 0.405·24-s + 1.89·25-s − 0.272·26-s + 0.784·27-s − 1.24·29-s + 1.37·30-s − 0.596·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 1.98T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 11 | \( 1 - 5.77T + 11T^{2} \) |
| 13 | \( 1 + 1.39T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 8.54T + 23T^{2} \) |
| 29 | \( 1 + 6.67T + 29T^{2} \) |
| 31 | \( 1 + 3.32T + 31T^{2} \) |
| 37 | \( 1 + 2.22T + 37T^{2} \) |
| 41 | \( 1 - 7.36T + 41T^{2} \) |
| 43 | \( 1 - 8.79T + 43T^{2} \) |
| 47 | \( 1 - 1.42T + 47T^{2} \) |
| 53 | \( 1 - 7.53T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 0.965T + 61T^{2} \) |
| 67 | \( 1 - 3.23T + 67T^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 83 | \( 1 + 0.899T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
| 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
−7.25712512700436353622556051570, −6.82628450704699900257099532476, −6.02534767175433638599526532589, −5.46677984469098550869504338909, −4.50142329094631920385180838662, −3.99803033382266221837212096994, −3.62369414274031428072525586510, −2.32151958290763182055347465059, −1.02725724216957291998703766789, 0,
1.02725724216957291998703766789, 2.32151958290763182055347465059, 3.62369414274031428072525586510, 3.99803033382266221837212096994, 4.50142329094631920385180838662, 5.46677984469098550869504338909, 6.02534767175433638599526532589, 6.82628450704699900257099532476, 7.25712512700436353622556051570
