| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 11-s + 12-s + 2·13-s + 14-s + 16-s − 2·17-s − 18-s − 4·19-s − 21-s + 22-s − 8·23-s − 24-s − 2·26-s + 27-s − 28-s + 6·29-s + 8·31-s − 32-s − 33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.218·21-s + 0.213·22-s − 1.66·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.174·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.440546575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440546575\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.44993232952611, −15.82141097166916, −15.41450349249809, −14.94455732823941, −13.98057817552620, −13.75815535287278, −13.02030931203033, −12.36536240036523, −11.86404020424334, −11.08584434692384, −10.45266244803394, −10.00348180408491, −9.443941239839298, −8.690362694599884, −8.167199510835164, −7.894772702991925, −6.765744419358355, −6.472172407672379, −5.767997150527426, −4.657532616864791, −4.050130519222306, −3.167972424347008, −2.448936378457324, −1.736528217685916, −0.5840917161895787,
0.5840917161895787, 1.736528217685916, 2.448936378457324, 3.167972424347008, 4.050130519222306, 4.657532616864791, 5.767997150527426, 6.472172407672379, 6.765744419358355, 7.894772702991925, 8.167199510835164, 8.690362694599884, 9.443941239839298, 10.00348180408491, 10.45266244803394, 11.08584434692384, 11.86404020424334, 12.36536240036523, 13.02030931203033, 13.75815535287278, 13.98057817552620, 14.94455732823941, 15.41450349249809, 15.82141097166916, 16.44993232952611
