| L(s) = 1 | + 3-s − 3.61·5-s + 0.982·7-s + 9-s − 3.24·11-s + 2.88·13-s − 3.61·15-s − 0.659·17-s + 2.91·19-s + 0.982·21-s − 1.79·23-s + 8.03·25-s + 27-s + 4.94·29-s + 1.83·31-s − 3.24·33-s − 3.54·35-s − 11.7·37-s + 2.88·39-s − 3.21·41-s − 7.99·43-s − 3.61·45-s + 6.45·47-s − 6.03·49-s − 0.659·51-s + 1.47·53-s + 11.7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.61·5-s + 0.371·7-s + 0.333·9-s − 0.978·11-s + 0.800·13-s − 0.932·15-s − 0.160·17-s + 0.668·19-s + 0.214·21-s − 0.374·23-s + 1.60·25-s + 0.192·27-s + 0.918·29-s + 0.330·31-s − 0.564·33-s − 0.599·35-s − 1.93·37-s + 0.462·39-s − 0.501·41-s − 1.21·43-s − 0.538·45-s + 0.941·47-s − 0.862·49-s − 0.0923·51-s + 0.202·53-s + 1.57·55-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 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 - 0.982T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 + 0.659T + 17T^{2} \) |
| 19 | \( 1 - 2.91T + 19T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 - 4.94T + 29T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 3.21T + 41T^{2} \) |
| 43 | \( 1 + 7.99T + 43T^{2} \) |
| 47 | \( 1 - 6.45T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 + 0.809T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 - 7.49T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 - 3.32T + 83T^{2} \) |
| 89 | \( 1 + 7.58T + 89T^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−7.894152508355267000821939494809, −7.18833626352020600292131824784, −6.56658264785904825767766857634, −5.32472165421513135289404596556, −4.77693043082299225068255602703, −3.83593856507465345901427494134, −3.40626078466208069477809746736, −2.49764327590567590828342408731, −1.26008824744431192658065527817, 0,
1.26008824744431192658065527817, 2.49764327590567590828342408731, 3.40626078466208069477809746736, 3.83593856507465345901427494134, 4.77693043082299225068255602703, 5.32472165421513135289404596556, 6.56658264785904825767766857634, 7.18833626352020600292131824784, 7.894152508355267000821939494809
