| L(s) = 1 | − 3·3-s + 18·5-s + 9·9-s + 36·11-s + 10·13-s − 54·15-s − 18·17-s + 100·19-s + 72·23-s + 199·25-s − 27·27-s − 234·29-s + 16·31-s − 108·33-s − 226·37-s − 30·39-s − 90·41-s + 452·43-s + 162·45-s − 432·47-s + 54·51-s + 414·53-s + 648·55-s − 300·57-s + 684·59-s − 422·61-s + 180·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.60·5-s + 1/3·9-s + 0.986·11-s + 0.213·13-s − 0.929·15-s − 0.256·17-s + 1.20·19-s + 0.652·23-s + 1.59·25-s − 0.192·27-s − 1.49·29-s + 0.0926·31-s − 0.569·33-s − 1.00·37-s − 0.123·39-s − 0.342·41-s + 1.60·43-s + 0.536·45-s − 1.34·47-s + 0.148·51-s + 1.07·53-s + 1.58·55-s − 0.697·57-s + 1.50·59-s − 0.885·61-s + 0.343·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.502025831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.502025831\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 - 16 T + p^{3} T^{2} \) |
| 37 | \( 1 + 226 T + p^{3} T^{2} \) |
| 41 | \( 1 + 90 T + p^{3} T^{2} \) |
| 43 | \( 1 - 452 T + p^{3} T^{2} \) |
| 47 | \( 1 + 432 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 684 T + p^{3} T^{2} \) |
| 61 | \( 1 + 422 T + p^{3} T^{2} \) |
| 67 | \( 1 - 332 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 26 T + p^{3} T^{2} \) |
| 79 | \( 1 - 512 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 630 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1054 T + p^{3} T^{2} \) |
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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
−10.21817075986854378394375965109, −9.441469866158331793668997144856, −8.910663576120968782704871338030, −7.35634645182343605326598318728, −6.50533169331643722300136046922, −5.73139632222972584336659735327, −4.99859964486060397057219143606, −3.56144558018467114887719862697, −2.06567960000741371147237745958, −1.05713204138579396975200055213,
1.05713204138579396975200055213, 2.06567960000741371147237745958, 3.56144558018467114887719862697, 4.99859964486060397057219143606, 5.73139632222972584336659735327, 6.50533169331643722300136046922, 7.35634645182343605326598318728, 8.910663576120968782704871338030, 9.441469866158331793668997144856, 10.21817075986854378394375965109
