| L(s) = 1 | + 2-s + 3-s + 4-s − 0.229·5-s + 6-s + 2.69·7-s + 8-s + 9-s − 0.229·10-s + 3.73·11-s + 12-s + 2.15·13-s + 2.69·14-s − 0.229·15-s + 16-s + 5.34·17-s + 18-s + 19-s − 0.229·20-s + 2.69·21-s + 3.73·22-s − 1.94·23-s + 24-s − 4.94·25-s + 2.15·26-s + 27-s + 2.69·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.102·5-s + 0.408·6-s + 1.01·7-s + 0.353·8-s + 0.333·9-s − 0.0726·10-s + 1.12·11-s + 0.288·12-s + 0.598·13-s + 0.719·14-s − 0.0592·15-s + 0.250·16-s + 1.29·17-s + 0.235·18-s + 0.229·19-s − 0.0513·20-s + 0.587·21-s + 0.796·22-s − 0.405·23-s + 0.204·24-s − 0.989·25-s + 0.423·26-s + 0.192·27-s + 0.508·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.335452637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.335452637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 0.229T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 - 2.15T + 13T^{2} \) |
| 17 | \( 1 - 5.34T + 17T^{2} \) |
| 23 | \( 1 + 1.94T + 23T^{2} \) |
| 29 | \( 1 + 5.72T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.91T + 37T^{2} \) |
| 41 | \( 1 + 4.50T + 41T^{2} \) |
| 43 | \( 1 - 9.41T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 4.80T + 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 + 2.77T + 73T^{2} \) |
| 79 | \( 1 - 9.77T + 79T^{2} \) |
| 83 | \( 1 + 7.84T + 83T^{2} \) |
| 89 | \( 1 + 3.87T + 89T^{2} \) |
| 97 | \( 1 + 6.33T + 97T^{2} \) |
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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
−7.87692689798358507226102659689, −7.56403977914151431538228171177, −6.56003847254220106637175940506, −5.89465410200709160507604750551, −5.14691269331630963661009325310, −4.25377340908372247047382373450, −3.76786125536740547023543340154, −2.96922074699199382525455840911, −1.81966775083111908662447333793, −1.22294653884838350817029885539,
1.22294653884838350817029885539, 1.81966775083111908662447333793, 2.96922074699199382525455840911, 3.76786125536740547023543340154, 4.25377340908372247047382373450, 5.14691269331630963661009325310, 5.89465410200709160507604750551, 6.56003847254220106637175940506, 7.56403977914151431538228171177, 7.87692689798358507226102659689
