| L(s) = 1 | − 4·2-s + 16·4-s + 12·5-s − 64·8-s − 48·10-s − 288·11-s − 737·13-s + 256·16-s − 156·17-s − 617·19-s + 192·20-s + 1.15e3·22-s − 4.59e3·23-s − 2.98e3·25-s + 2.94e3·26-s + 5.30e3·29-s − 2.51e3·31-s − 1.02e3·32-s + 624·34-s + 2.37e3·37-s + 2.46e3·38-s − 768·40-s − 1.42e4·41-s − 1.57e3·43-s − 4.60e3·44-s + 1.83e4·46-s + 1.72e4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.214·5-s − 0.353·8-s − 0.151·10-s − 0.717·11-s − 1.20·13-s + 1/4·16-s − 0.130·17-s − 0.392·19-s + 0.107·20-s + 0.507·22-s − 1.81·23-s − 0.953·25-s + 0.855·26-s + 1.17·29-s − 0.469·31-s − 0.176·32-s + 0.0925·34-s + 0.285·37-s + 0.277·38-s − 0.0758·40-s − 1.32·41-s − 0.130·43-s − 0.358·44-s + 1.28·46-s + 1.14·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7540042991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7540042991\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 12 T + p^{5} T^{2} \) |
| 11 | \( 1 + 288 T + p^{5} T^{2} \) |
| 13 | \( 1 + 737 T + p^{5} T^{2} \) |
| 17 | \( 1 + 156 T + p^{5} T^{2} \) |
| 19 | \( 1 + 617 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4596 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5304 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2513 T + p^{5} T^{2} \) |
| 37 | \( 1 - 2375 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14280 T + p^{5} T^{2} \) |
| 43 | \( 1 + 1579 T + p^{5} T^{2} \) |
| 47 | \( 1 - 17268 T + p^{5} T^{2} \) |
| 53 | \( 1 - 18612 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28428 T + p^{5} T^{2} \) |
| 61 | \( 1 + 15566 T + p^{5} T^{2} \) |
| 67 | \( 1 + 8053 T + p^{5} T^{2} \) |
| 71 | \( 1 + 13020 T + p^{5} T^{2} \) |
| 73 | \( 1 - 50263 T + p^{5} T^{2} \) |
| 79 | \( 1 - 30155 T + p^{5} T^{2} \) |
| 83 | \( 1 - 99276 T + p^{5} T^{2} \) |
| 89 | \( 1 + 52104 T + p^{5} T^{2} \) |
| 97 | \( 1 + 116222 T + p^{5} T^{2} \) |
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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
−9.566618031920247567718798265300, −8.456082409526653507856416202312, −7.84305397562213467701776833876, −6.99162636574207428339691611129, −6.02820126389137101147370890774, −5.12089313603109923458296763406, −3.96970296140917622686554993354, −2.59422144037968830282454713825, −1.91827735482604998162427756769, −0.40941798024380133219812168450,
0.40941798024380133219812168450, 1.91827735482604998162427756769, 2.59422144037968830282454713825, 3.96970296140917622686554993354, 5.12089313603109923458296763406, 6.02820126389137101147370890774, 6.99162636574207428339691611129, 7.84305397562213467701776833876, 8.456082409526653507856416202312, 9.566618031920247567718798265300
