| L(s) = 1 | + 3-s + 5-s + 2.57·7-s + 9-s − 5.06·11-s − 1.30·13-s + 15-s − 4.52·17-s − 0.372·19-s + 2.57·21-s − 6.09·23-s + 25-s + 27-s − 4.59·29-s − 8.38·31-s − 5.06·33-s + 2.57·35-s − 8.09·37-s − 1.30·39-s − 5.44·41-s + 9.10·43-s + 45-s − 6.61·47-s − 0.372·49-s − 4.52·51-s + 0.143·53-s − 5.06·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.973·7-s + 0.333·9-s − 1.52·11-s − 0.361·13-s + 0.258·15-s − 1.09·17-s − 0.0855·19-s + 0.561·21-s − 1.27·23-s + 0.200·25-s + 0.192·27-s − 0.853·29-s − 1.50·31-s − 0.881·33-s + 0.435·35-s − 1.33·37-s − 0.208·39-s − 0.850·41-s + 1.38·43-s + 0.149·45-s − 0.964·47-s − 0.0532·49-s − 0.633·51-s + 0.0197·53-s − 0.682·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 - 2.57T + 7T^{2} \) |
| 11 | \( 1 + 5.06T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 + 0.372T + 19T^{2} \) |
| 23 | \( 1 + 6.09T + 23T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 + 8.38T + 31T^{2} \) |
| 37 | \( 1 + 8.09T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 - 9.10T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 - 0.143T + 53T^{2} \) |
| 59 | \( 1 + 9.34T + 59T^{2} \) |
| 61 | \( 1 - 6.21T + 61T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 - 4.42T + 79T^{2} \) |
| 83 | \( 1 - 0.605T + 83T^{2} \) |
| 89 | \( 1 + 7.85T + 89T^{2} \) |
| 97 | \( 1 + 6.62T + 97T^{2} \) |
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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
−8.084917335312744342145653370994, −7.52822419363005259856138543150, −6.75936150069824526650746922968, −5.65021610493869077190055186650, −5.11900040174032722638050284979, −4.32904430336698716458593396942, −3.33635126198124724720516721769, −2.17967035703516148307993723165, −1.89446385302265457688205461532, 0,
1.89446385302265457688205461532, 2.17967035703516148307993723165, 3.33635126198124724720516721769, 4.32904430336698716458593396942, 5.11900040174032722638050284979, 5.65021610493869077190055186650, 6.75936150069824526650746922968, 7.52822419363005259856138543150, 8.084917335312744342145653370994
