| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 7-s + 8-s + 9-s + 2·10-s + 4·11-s + 12-s + 14-s + 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s + 2·20-s + 21-s + 4·22-s − 4·23-s + 24-s − 25-s + 27-s + 28-s − 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.806235520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.806235520\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.79676275326953184687961611093, −7.19540786561249063152549847427, −6.39876551984950476041029050521, −5.84484580912043443889146565919, −5.12877355795531698006926879996, −4.18426885950219780500736531500, −3.73509942720132211562206816978, −2.66226460338892757420373056267, −1.97545120936719195855762021342, −1.17935276529577646286253717029,
1.17935276529577646286253717029, 1.97545120936719195855762021342, 2.66226460338892757420373056267, 3.73509942720132211562206816978, 4.18426885950219780500736531500, 5.12877355795531698006926879996, 5.84484580912043443889146565919, 6.39876551984950476041029050521, 7.19540786561249063152549847427, 7.79676275326953184687961611093
