| L(s) = 1 | − 2-s + 4-s + 2.66·5-s − 3.18·7-s − 8-s − 2.66·10-s + 5.51·11-s + 2.64·13-s + 3.18·14-s + 16-s − 3.74·17-s − 6.37·19-s + 2.66·20-s − 5.51·22-s + 2.12·25-s − 2.64·26-s − 3.18·28-s + 5.50·29-s − 10.6·31-s − 32-s + 3.74·34-s − 8.50·35-s + 9.36·37-s + 6.37·38-s − 2.66·40-s + 0.424·41-s − 5.38·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.19·5-s − 1.20·7-s − 0.353·8-s − 0.844·10-s + 1.66·11-s + 0.734·13-s + 0.851·14-s + 0.250·16-s − 0.908·17-s − 1.46·19-s + 0.596·20-s − 1.17·22-s + 0.425·25-s − 0.519·26-s − 0.602·28-s + 1.02·29-s − 1.91·31-s − 0.176·32-s + 0.642·34-s − 1.43·35-s + 1.53·37-s + 1.03·38-s − 0.422·40-s + 0.0663·41-s − 0.820·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 2.66T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 29 | \( 1 - 5.50T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 - 9.36T + 37T^{2} \) |
| 41 | \( 1 - 0.424T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 + 9.38T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 5.64T + 59T^{2} \) |
| 61 | \( 1 + 7.39T + 61T^{2} \) |
| 67 | \( 1 + 4.30T + 67T^{2} \) |
| 71 | \( 1 + 1.91T + 71T^{2} \) |
| 73 | \( 1 + 5.37T + 73T^{2} \) |
| 79 | \( 1 - 6.61T + 79T^{2} \) |
| 83 | \( 1 + 5.15T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
| 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
−7.11628956219437910151408236476, −6.44432366797090381077079154869, −6.35841749379897881482247107287, −5.69113966302674265306304495093, −4.42499923329713229560379208365, −3.76028195762722457579948551360, −2.85970411739494850604020985321, −1.97401819198881136210101145532, −1.31873950076176200310164566330, 0,
1.31873950076176200310164566330, 1.97401819198881136210101145532, 2.85970411739494850604020985321, 3.76028195762722457579948551360, 4.42499923329713229560379208365, 5.69113966302674265306304495093, 6.35841749379897881482247107287, 6.44432366797090381077079154869, 7.11628956219437910151408236476
