| L(s) = 1 | + 2-s − 2.96·3-s + 4-s − 5-s − 2.96·6-s − 3.63·7-s + 8-s + 5.78·9-s − 10-s − 11-s − 2.96·12-s − 5.70·13-s − 3.63·14-s + 2.96·15-s + 16-s + 4.06·17-s + 5.78·18-s − 1.71·19-s − 20-s + 10.7·21-s − 22-s + 7.48·23-s − 2.96·24-s + 25-s − 5.70·26-s − 8.26·27-s − 3.63·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.71·3-s + 0.5·4-s − 0.447·5-s − 1.21·6-s − 1.37·7-s + 0.353·8-s + 1.92·9-s − 0.316·10-s − 0.301·11-s − 0.855·12-s − 1.58·13-s − 0.972·14-s + 0.765·15-s + 0.250·16-s + 0.985·17-s + 1.36·18-s − 0.393·19-s − 0.223·20-s + 2.35·21-s − 0.213·22-s + 1.56·23-s − 0.605·24-s + 0.200·25-s − 1.11·26-s − 1.59·27-s − 0.687·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 + 2.96T + 3T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 - 4.06T + 17T^{2} \) |
| 19 | \( 1 + 1.71T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 + 6.31T + 29T^{2} \) |
| 31 | \( 1 + 3.36T + 31T^{2} \) |
| 37 | \( 1 - 8.63T + 37T^{2} \) |
| 41 | \( 1 + 0.774T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 2.15T + 59T^{2} \) |
| 61 | \( 1 - 5.33T + 61T^{2} \) |
| 67 | \( 1 + 3.59T + 67T^{2} \) |
| 71 | \( 1 - 0.474T + 71T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 1.13T + 83T^{2} \) |
| 89 | \( 1 - 8.12T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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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
−7.19144674030959152555777293656, −6.76373730326195749578227483711, −5.85518501769136645638011407197, −5.51760161522703291927937026665, −4.81139377491087189164039646692, −4.10949137659506245088987898786, −3.24731175646405330242822750019, −2.41193758509946526557762752393, −0.909214394337603863933577435139, 0,
0.909214394337603863933577435139, 2.41193758509946526557762752393, 3.24731175646405330242822750019, 4.10949137659506245088987898786, 4.81139377491087189164039646692, 5.51760161522703291927937026665, 5.85518501769136645638011407197, 6.76373730326195749578227483711, 7.19144674030959152555777293656
