| L(s) = 1 | + 3-s + 0.406·5-s − 3.65·7-s + 9-s + 2.14·11-s + 3.44·13-s + 0.406·15-s − 3.83·17-s − 3.34·19-s − 3.65·21-s − 3.64·23-s − 4.83·25-s + 27-s + 9.24·29-s + 1.08·31-s + 2.14·33-s − 1.48·35-s − 2.54·37-s + 3.44·39-s − 8.54·41-s + 9.55·43-s + 0.406·45-s + 0.898·47-s + 6.38·49-s − 3.83·51-s − 1.91·53-s + 0.871·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.181·5-s − 1.38·7-s + 0.333·9-s + 0.646·11-s + 0.955·13-s + 0.104·15-s − 0.929·17-s − 0.767·19-s − 0.798·21-s − 0.760·23-s − 0.966·25-s + 0.192·27-s + 1.71·29-s + 0.194·31-s + 0.373·33-s − 0.251·35-s − 0.418·37-s + 0.551·39-s − 1.33·41-s + 1.45·43-s + 0.0605·45-s + 0.131·47-s + 0.911·49-s − 0.536·51-s − 0.262·53-s + 0.117·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 - 0.406T + 5T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 - 2.14T + 11T^{2} \) |
| 13 | \( 1 - 3.44T + 13T^{2} \) |
| 17 | \( 1 + 3.83T + 17T^{2} \) |
| 19 | \( 1 + 3.34T + 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 - 9.24T + 29T^{2} \) |
| 31 | \( 1 - 1.08T + 31T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 - 9.55T + 43T^{2} \) |
| 47 | \( 1 - 0.898T + 47T^{2} \) |
| 53 | \( 1 + 1.91T + 53T^{2} \) |
| 59 | \( 1 - 0.219T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 7.20T + 67T^{2} \) |
| 71 | \( 1 + 7.59T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 2.82T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 5.24T + 89T^{2} \) |
| 97 | \( 1 - 3.08T + 97T^{2} \) |
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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
−7.78348332406853455094993778061, −6.83681231059433229618496608568, −6.32684781246954518387949832461, −5.93307871948103169704627705363, −4.55727301697539612436441520785, −3.96113387116352486363117327003, −3.23633510550616128290535132741, −2.43288863216256088132689546784, −1.42786930227913356198329182258, 0,
1.42786930227913356198329182258, 2.43288863216256088132689546784, 3.23633510550616128290535132741, 3.96113387116352486363117327003, 4.55727301697539612436441520785, 5.93307871948103169704627705363, 6.32684781246954518387949832461, 6.83681231059433229618496608568, 7.78348332406853455094993778061
