| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 7·19-s + 21-s − 3·23-s + 24-s + 26-s − 27-s − 28-s + 6·29-s + 5·31-s − 32-s + 34-s + 36-s − 11·37-s − 7·38-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 − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.60·19-s + 0.218·21-s − 0.625·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.898·31-s − 0.176·32-s + 0.171·34-s + 1/6·36-s − 1.80·37-s − 1.13·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.616443043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.616443043\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.44183878217511, −11.98870024216561, −11.80088361957613, −11.39959854658609, −10.69372113610038, −10.27726104881474, −9.890098956923253, −9.723178514082552, −8.896078158204750, −8.588805985938599, −8.098241824015182, −7.472746758505483, −7.047811534355017, −6.718265546121751, −6.156811412586605, −5.653623041737528, −5.104659381897248, −4.735905740074892, −3.986736894470464, −3.276126322683540, −3.056296852741795, −2.117444263493626, −1.743037155020108, −0.8023835962789829, −0.5382192065899512,
0.5382192065899512, 0.8023835962789829, 1.743037155020108, 2.117444263493626, 3.056296852741795, 3.276126322683540, 3.986736894470464, 4.735905740074892, 5.104659381897248, 5.653623041737528, 6.156811412586605, 6.718265546121751, 7.047811534355017, 7.472746758505483, 8.098241824015182, 8.588805985938599, 8.896078158204750, 9.723178514082552, 9.890098956923253, 10.27726104881474, 10.69372113610038, 11.39959854658609, 11.80088361957613, 11.98870024216561, 12.44183878217511
