| L(s) = 1 | + 3.10·3-s − 5-s + 2.28·7-s + 6.62·9-s + 4.72·11-s − 5.33·13-s − 3.10·15-s + 4.52·17-s + 3.91·19-s + 7.10·21-s − 6.52·23-s + 25-s + 11.2·27-s + 9.59·29-s − 6.44·31-s + 14.6·33-s − 2.28·35-s − 3.94·37-s − 16.5·39-s − 3.54·41-s − 43-s − 6.62·45-s + 3.10·47-s − 1.75·49-s + 14.0·51-s + 12.7·53-s − 4.72·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s − 0.447·5-s + 0.865·7-s + 2.20·9-s + 1.42·11-s − 1.48·13-s − 0.801·15-s + 1.09·17-s + 0.898·19-s + 1.54·21-s − 1.36·23-s + 0.200·25-s + 2.16·27-s + 1.78·29-s − 1.15·31-s + 2.55·33-s − 0.386·35-s − 0.648·37-s − 2.65·39-s − 0.553·41-s − 0.152·43-s − 0.987·45-s + 0.452·47-s − 0.251·49-s + 1.96·51-s + 1.75·53-s − 0.637·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.146079976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.146079976\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 13 | \( 1 + 5.33T + 13T^{2} \) |
| 17 | \( 1 - 4.52T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 - 9.59T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 + 3.54T + 41T^{2} \) |
| 47 | \( 1 - 3.10T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 9.04T + 59T^{2} \) |
| 61 | \( 1 + 3.97T + 61T^{2} \) |
| 67 | \( 1 - 0.980T + 67T^{2} \) |
| 71 | \( 1 - 8.72T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 0.235T + 79T^{2} \) |
| 83 | \( 1 + 6.78T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 3.94T + 97T^{2} \) |
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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
−8.463434143018545291485756843529, −7.958345991697513683999787738352, −7.37665462423464673201501820061, −6.76171809541773881932063114856, −5.36714473844431124323786300714, −4.47664722984129794215740635018, −3.80625102265320783654569240317, −3.07182877708277766918821340199, −2.10034142162033598549283707377, −1.25569029819167276975254820073,
1.25569029819167276975254820073, 2.10034142162033598549283707377, 3.07182877708277766918821340199, 3.80625102265320783654569240317, 4.47664722984129794215740635018, 5.36714473844431124323786300714, 6.76171809541773881932063114856, 7.37665462423464673201501820061, 7.958345991697513683999787738352, 8.463434143018545291485756843529
