| L(s) = 1 | + 3-s − 2.29·5-s − 7-s + 9-s − 4.29·11-s − 3.87·13-s − 2.29·15-s − 17-s − 3.08·19-s − 21-s + 3.08·23-s + 0.276·25-s + 27-s + 6·29-s + 6.16·31-s − 4.29·33-s + 2.29·35-s + 0.786·37-s − 3.87·39-s − 1.08·41-s − 5.87·43-s − 2.29·45-s − 6.16·47-s + 49-s − 51-s − 2.95·53-s + 9.87·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.02·5-s − 0.377·7-s + 0.333·9-s − 1.29·11-s − 1.07·13-s − 0.593·15-s − 0.242·17-s − 0.707·19-s − 0.218·21-s + 0.643·23-s + 0.0553·25-s + 0.192·27-s + 1.11·29-s + 1.10·31-s − 0.748·33-s + 0.388·35-s + 0.129·37-s − 0.619·39-s − 0.169·41-s − 0.895·43-s − 0.342·45-s − 0.899·47-s + 0.142·49-s − 0.140·51-s − 0.405·53-s + 1.33·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.072771913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.072771913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2.29T + 5T^{2} \) |
| 11 | \( 1 + 4.29T + 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 19 | \( 1 + 3.08T + 19T^{2} \) |
| 23 | \( 1 - 3.08T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 6.16T + 31T^{2} \) |
| 37 | \( 1 - 0.786T + 37T^{2} \) |
| 41 | \( 1 + 1.08T + 41T^{2} \) |
| 43 | \( 1 + 5.87T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 + 2.95T + 53T^{2} \) |
| 59 | \( 1 - 7.02T + 59T^{2} \) |
| 61 | \( 1 + 5.02T + 61T^{2} \) |
| 67 | \( 1 + 4.95T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 1.38T + 73T^{2} \) |
| 79 | \( 1 - 2.78T + 79T^{2} \) |
| 83 | \( 1 - 4.36T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.089741203578080816260973777388, −7.55563241897776195663107974392, −6.89260712453926549664717789150, −6.10822121568372702479257103101, −4.79805057991774791586804124146, −4.69395634370256412208224015291, −3.49915208998746484028128010957, −2.88347758299567771910933028806, −2.12218887990436832979342738792, −0.50362081728573779615504243150,
0.50362081728573779615504243150, 2.12218887990436832979342738792, 2.88347758299567771910933028806, 3.49915208998746484028128010957, 4.69395634370256412208224015291, 4.79805057991774791586804124146, 6.10822121568372702479257103101, 6.89260712453926549664717789150, 7.55563241897776195663107974392, 8.089741203578080816260973777388
