| L(s) = 1 | − 2.20·2-s − 3-s + 2.85·4-s − 5-s + 2.20·6-s − 1.89·8-s + 9-s + 2.20·10-s − 11-s − 2.85·12-s + 4.37·13-s + 15-s − 1.54·16-s + 1.68·17-s − 2.20·18-s − 3.68·19-s − 2.85·20-s + 2.20·22-s − 8.69·23-s + 1.89·24-s + 25-s − 9.65·26-s − 27-s + 9.97·29-s − 2.20·30-s + 0.0540·31-s + 7.19·32-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 0.577·3-s + 1.42·4-s − 0.447·5-s + 0.899·6-s − 0.669·8-s + 0.333·9-s + 0.697·10-s − 0.301·11-s − 0.825·12-s + 1.21·13-s + 0.258·15-s − 0.385·16-s + 0.409·17-s − 0.519·18-s − 0.846·19-s − 0.639·20-s + 0.469·22-s − 1.81·23-s + 0.386·24-s + 0.200·25-s − 1.89·26-s − 0.192·27-s + 1.85·29-s − 0.402·30-s + 0.00969·31-s + 1.27·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 - 1.68T + 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 23 | \( 1 + 8.69T + 23T^{2} \) |
| 29 | \( 1 - 9.97T + 29T^{2} \) |
| 31 | \( 1 - 0.0540T + 31T^{2} \) |
| 37 | \( 1 - 5.34T + 37T^{2} \) |
| 41 | \( 1 + 3.15T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 4.04T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 8.59T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 6.60T + 67T^{2} \) |
| 71 | \( 1 + 3.96T + 71T^{2} \) |
| 73 | \( 1 + 3.42T + 73T^{2} \) |
| 79 | \( 1 - 1.95T + 79T^{2} \) |
| 83 | \( 1 + 4.53T + 83T^{2} \) |
| 89 | \( 1 - 4.88T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
| 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
−7.890057718723258777048593012664, −6.80210535981743345500218106856, −6.40348893720600151176265239494, −5.68031108846203355880872660033, −4.57257577401445158616505276936, −3.96487746567359605907720803187, −2.83603259012384341381341688796, −1.80967123720913936600640543958, −0.964598488763526176318104120214, 0,
0.964598488763526176318104120214, 1.80967123720913936600640543958, 2.83603259012384341381341688796, 3.96487746567359605907720803187, 4.57257577401445158616505276936, 5.68031108846203355880872660033, 6.40348893720600151176265239494, 6.80210535981743345500218106856, 7.890057718723258777048593012664
