| L(s) = 1 | − 2-s − 0.257·3-s + 4-s + 0.105·5-s + 0.257·6-s − 1.45·7-s − 8-s − 2.93·9-s − 0.105·10-s − 11-s − 0.257·12-s − 3.63·13-s + 1.45·14-s − 0.0271·15-s + 16-s − 4.13·17-s + 2.93·18-s − 1.69·19-s + 0.105·20-s + 0.376·21-s + 22-s − 3.69·23-s + 0.257·24-s − 4.98·25-s + 3.63·26-s + 1.53·27-s − 1.45·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.148·3-s + 0.5·4-s + 0.0471·5-s + 0.105·6-s − 0.551·7-s − 0.353·8-s − 0.977·9-s − 0.0333·10-s − 0.301·11-s − 0.0744·12-s − 1.00·13-s + 0.390·14-s − 0.00701·15-s + 0.250·16-s − 1.00·17-s + 0.691·18-s − 0.388·19-s + 0.0235·20-s + 0.0821·21-s + 0.213·22-s − 0.771·23-s + 0.0526·24-s − 0.997·25-s + 0.712·26-s + 0.294·27-s − 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4202266741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4202266741\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 0.257T + 3T^{2} \) |
| 5 | \( 1 - 0.105T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 13 | \( 1 + 3.63T + 13T^{2} \) |
| 17 | \( 1 + 4.13T + 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 + 3.69T + 23T^{2} \) |
| 29 | \( 1 + 0.761T + 29T^{2} \) |
| 31 | \( 1 + 2.13T + 31T^{2} \) |
| 37 | \( 1 - 5.15T + 37T^{2} \) |
| 41 | \( 1 + 1.86T + 41T^{2} \) |
| 43 | \( 1 - 9.03T + 43T^{2} \) |
| 47 | \( 1 - 0.00354T + 47T^{2} \) |
| 53 | \( 1 + 5.66T + 53T^{2} \) |
| 59 | \( 1 - 4.58T + 59T^{2} \) |
| 61 | \( 1 - 6.62T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 4.63T + 73T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 - 4.80T + 83T^{2} \) |
| 89 | \( 1 - 5.15T + 89T^{2} \) |
| 97 | \( 1 + 3.95T + 97T^{2} \) |
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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
−8.344306275048235294914470284047, −7.78266346926441956962249008720, −6.97141300037493736795683957524, −6.22139975724043727348055703442, −5.65668423747146272342979330572, −4.68661336119449614313476472022, −3.68618169209011814154057161689, −2.62534422950393994219698199905, −2.07896491131877574932565183651, −0.37810722504315831305335233092,
0.37810722504315831305335233092, 2.07896491131877574932565183651, 2.62534422950393994219698199905, 3.68618169209011814154057161689, 4.68661336119449614313476472022, 5.65668423747146272342979330572, 6.22139975724043727348055703442, 6.97141300037493736795683957524, 7.78266346926441956962249008720, 8.344306275048235294914470284047
