| L(s) = 1 | + 2.64·5-s − 2.64·11-s − 2·13-s + 19-s − 7.93·23-s + 2.00·25-s + 5.29·29-s − 7·31-s − 3·37-s + 7.93·41-s − 2·43-s − 5.29·47-s − 10.5·53-s − 7.00·55-s − 5.29·59-s + 8·61-s − 5.29·65-s − 2·67-s − 2.64·71-s − 10·73-s + 10·79-s − 15.8·83-s − 2.64·89-s + 2.64·95-s − 16·97-s + 10.5·101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.18·5-s − 0.797·11-s − 0.554·13-s + 0.229·19-s − 1.65·23-s + 0.400·25-s + 0.982·29-s − 1.25·31-s − 0.493·37-s + 1.23·41-s − 0.304·43-s − 0.771·47-s − 1.45·53-s − 0.943·55-s − 0.688·59-s + 1.02·61-s − 0.656·65-s − 0.244·67-s − 0.313·71-s − 1.17·73-s + 1.12·79-s − 1.74·83-s − 0.280·89-s + 0.271·95-s − 1.62·97-s + 1.05·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.64T + 5T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 7.93T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 7.93T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 5.29T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 2.64T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 2.64T + 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−7.85360324628915025379674596154, −7.12657971276713424679338996987, −6.24092616945909861241109926079, −5.70672700818231930119911797703, −5.05564438735964791054919015731, −4.20907481336523939100022583722, −3.09607832012245094879726873730, −2.29733670190307766320676724043, −1.57269289143205613462870078143, 0,
1.57269289143205613462870078143, 2.29733670190307766320676724043, 3.09607832012245094879726873730, 4.20907481336523939100022583722, 5.05564438735964791054919015731, 5.70672700818231930119911797703, 6.24092616945909861241109926079, 7.12657971276713424679338996987, 7.85360324628915025379674596154
