| L(s) = 1 | + 1.71·2-s − 1.85·3-s + 0.930·4-s − 5-s − 3.18·6-s + 0.899·7-s − 1.83·8-s + 0.457·9-s − 1.71·10-s + 2.44·11-s − 1.73·12-s − 0.209·13-s + 1.54·14-s + 1.85·15-s − 4.99·16-s − 1.71·17-s + 0.783·18-s + 5.63·19-s − 0.930·20-s − 1.67·21-s + 4.18·22-s + 2.43·23-s + 3.40·24-s + 25-s − 0.359·26-s + 4.72·27-s + 0.837·28-s + ⋯ |
| L(s) = 1 | + 1.21·2-s − 1.07·3-s + 0.465·4-s − 0.447·5-s − 1.29·6-s + 0.340·7-s − 0.647·8-s + 0.152·9-s − 0.541·10-s + 0.736·11-s − 0.499·12-s − 0.0582·13-s + 0.411·14-s + 0.480·15-s − 1.24·16-s − 0.415·17-s + 0.184·18-s + 1.29·19-s − 0.208·20-s − 0.365·21-s + 0.891·22-s + 0.507·23-s + 0.694·24-s + 0.200·25-s − 0.0704·26-s + 0.909·27-s + 0.158·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 1.71T + 2T^{2} \) |
| 3 | \( 1 + 1.85T + 3T^{2} \) |
| 7 | \( 1 - 0.899T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 0.209T + 13T^{2} \) |
| 17 | \( 1 + 1.71T + 17T^{2} \) |
| 19 | \( 1 - 5.63T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 31 | \( 1 - 1.99T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 + 0.676T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 4.91T + 53T^{2} \) |
| 59 | \( 1 + 6.85T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 9.83T + 71T^{2} \) |
| 73 | \( 1 + 1.68T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 7.14T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.87948838540617165081457111889, −6.88085317957797393008668990194, −6.44781749967323611853729814063, −5.57233640350281732761844915734, −5.05593450187079298606649457961, −4.47106500014850836984026237695, −3.58643984957020234964551952557, −2.84799322170638136464188460136, −1.34128182371658093219847006187, 0,
1.34128182371658093219847006187, 2.84799322170638136464188460136, 3.58643984957020234964551952557, 4.47106500014850836984026237695, 5.05593450187079298606649457961, 5.57233640350281732761844915734, 6.44781749967323611853729814063, 6.88085317957797393008668990194, 7.87948838540617165081457111889
