| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s + 2·11-s − 12-s − 2·13-s + 4·14-s + 16-s − 18-s − 2·19-s + 4·21-s − 2·22-s + 24-s + 2·26-s − 27-s − 4·28-s − 6·29-s + 4·31-s − 32-s − 2·33-s + 36-s − 2·37-s + 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.872·21-s − 0.426·22-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.348·33-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2594696730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2594696730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.90545088456527, −13.32522008877028, −13.00436564497648, −12.37215335586649, −11.97378755389060, −11.57918276003325, −10.93884973263419, −10.32742947887802, −9.957329973160529, −9.633535040906980, −9.005556228454877, −8.586326727599406, −7.892064363919797, −7.157273002380967, −6.818348648232361, −6.448308847460671, −5.835713567364836, −5.312174600934255, −4.555431503358240, −3.779604060021773, −3.378916419605781, −2.602385447471205, −1.932466121446714, −1.113320171109145, −0.2114622060786561,
0.2114622060786561, 1.113320171109145, 1.932466121446714, 2.602385447471205, 3.378916419605781, 3.779604060021773, 4.555431503358240, 5.312174600934255, 5.835713567364836, 6.448308847460671, 6.818348648232361, 7.157273002380967, 7.892064363919797, 8.586326727599406, 9.005556228454877, 9.633535040906980, 9.957329973160529, 10.32742947887802, 10.93884973263419, 11.57918276003325, 11.97378755389060, 12.37215335586649, 13.00436564497648, 13.32522008877028, 13.90545088456527
