| L(s) = 1 | − 3-s − 2·9-s + 11-s + 2·13-s + 4·19-s + 5·23-s + 5·27-s − 29-s + 10·31-s − 33-s + 2·37-s − 2·39-s − 11·41-s − 5·43-s − 6·53-s − 4·57-s + 10·59-s − 11·61-s − 3·67-s − 5·69-s + 12·71-s + 8·73-s + 14·79-s + 81-s − 5·83-s + 87-s − 89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.301·11-s + 0.554·13-s + 0.917·19-s + 1.04·23-s + 0.962·27-s − 0.185·29-s + 1.79·31-s − 0.174·33-s + 0.328·37-s − 0.320·39-s − 1.71·41-s − 0.762·43-s − 0.824·53-s − 0.529·57-s + 1.30·59-s − 1.40·61-s − 0.366·67-s − 0.601·69-s + 1.42·71-s + 0.936·73-s + 1.57·79-s + 1/9·81-s − 0.548·83-s + 0.107·87-s − 0.105·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.057104564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.057104564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.71113213329782, −13.34091650473680, −12.54301860827330, −12.16191949509423, −11.66251771308320, −11.31025927621481, −10.87522229872342, −10.30474246884454, −9.752375807863163, −9.298636000080151, −8.657987222456848, −8.262910850518784, −7.790851190419781, −6.969180869594754, −6.558548096761671, −6.205482411939310, −5.437348956195300, −5.095197722214815, −4.594068867914247, −3.770222495805694, −3.176328866096248, −2.786367818790067, −1.841799007345171, −1.097771497553377, −0.5311769017906902,
0.5311769017906902, 1.097771497553377, 1.841799007345171, 2.786367818790067, 3.176328866096248, 3.770222495805694, 4.594068867914247, 5.095197722214815, 5.437348956195300, 6.205482411939310, 6.558548096761671, 6.969180869594754, 7.790851190419781, 8.262910850518784, 8.657987222456848, 9.298636000080151, 9.752375807863163, 10.30474246884454, 10.87522229872342, 11.31025927621481, 11.66251771308320, 12.16191949509423, 12.54301860827330, 13.34091650473680, 13.71113213329782
