| L(s) = 1 | + 2·3-s + 2·5-s + 7-s + 9-s + 2·11-s + 4·13-s + 4·15-s − 2·17-s + 2·21-s + 23-s − 25-s − 4·27-s − 2·29-s + 4·33-s + 2·35-s − 4·37-s + 8·39-s + 6·41-s + 2·43-s + 2·45-s + 4·47-s + 49-s − 4·51-s + 4·55-s + 2·59-s − 10·61-s + 63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 1.03·15-s − 0.485·17-s + 0.436·21-s + 0.208·23-s − 1/5·25-s − 0.769·27-s − 0.371·29-s + 0.696·33-s + 0.338·35-s − 0.657·37-s + 1.28·39-s + 0.937·41-s + 0.304·43-s + 0.298·45-s + 0.583·47-s + 1/7·49-s − 0.560·51-s + 0.539·55-s + 0.260·59-s − 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.076717033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.076717033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−9.285698479120796413148588078333, −9.016917703525192158700193739397, −8.219999410090880743771516627849, −7.35688181088306878184242960478, −6.29332429532612070908675608623, −5.60861667256053636796846429014, −4.31570327029482052943750750509, −3.43973210784164728578386989669, −2.36115174393216724794363682538, −1.47018544701661058451472686608,
1.47018544701661058451472686608, 2.36115174393216724794363682538, 3.43973210784164728578386989669, 4.31570327029482052943750750509, 5.60861667256053636796846429014, 6.29332429532612070908675608623, 7.35688181088306878184242960478, 8.219999410090880743771516627849, 9.016917703525192158700193739397, 9.285698479120796413148588078333
