| L(s) = 1 | + 4·2-s + 16·4-s + 24·5-s + 77·7-s + 64·8-s + 96·10-s + 408·11-s + 89·13-s + 308·14-s + 256·16-s + 2.08e3·17-s − 2.61e3·19-s + 384·20-s + 1.63e3·22-s + 1.75e3·23-s − 2.54e3·25-s + 356·26-s + 1.23e3·28-s − 7.29e3·29-s + 2.34e3·31-s + 1.02e3·32-s + 8.35e3·34-s + 1.84e3·35-s − 4.99e3·37-s − 1.04e4·38-s + 1.53e3·40-s − 6.52e3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.429·5-s + 0.593·7-s + 0.353·8-s + 0.303·10-s + 1.01·11-s + 0.146·13-s + 0.419·14-s + 1/4·16-s + 1.75·17-s − 1.66·19-s + 0.214·20-s + 0.718·22-s + 0.690·23-s − 0.815·25-s + 0.103·26-s + 0.296·28-s − 1.61·29-s + 0.438·31-s + 0.176·32-s + 1.23·34-s + 0.254·35-s − 0.599·37-s − 1.17·38-s + 0.151·40-s − 0.606·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.929379865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.929379865\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 24 T + p^{5} T^{2} \) |
| 7 | \( 1 - 11 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 408 T + p^{5} T^{2} \) |
| 13 | \( 1 - 89 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2088 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2617 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1752 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7296 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2348 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4993 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6528 T + p^{5} T^{2} \) |
| 43 | \( 1 + 6232 T + p^{5} T^{2} \) |
| 47 | \( 1 + 29832 T + p^{5} T^{2} \) |
| 53 | \( 1 - 22608 T + p^{5} T^{2} \) |
| 59 | \( 1 - 19608 T + p^{5} T^{2} \) |
| 61 | \( 1 + 22045 T + p^{5} T^{2} \) |
| 67 | \( 1 - 48131 T + p^{5} T^{2} \) |
| 71 | \( 1 - 720 p T + p^{5} T^{2} \) |
| 73 | \( 1 - 30737 T + p^{5} T^{2} \) |
| 79 | \( 1 - 38219 T + p^{5} T^{2} \) |
| 83 | \( 1 - 8112 T + p^{5} T^{2} \) |
| 89 | \( 1 + 44280 T + p^{5} T^{2} \) |
| 97 | \( 1 + 136651 T + p^{5} T^{2} \) |
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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
−14.44794685247518163269796332226, −13.28979651028398063143819666796, −12.14473325487750635672888951581, −11.09064865224903604686586560012, −9.733394579255475606520879201282, −8.174556655790753703874742506761, −6.61245659144104597163627425129, −5.31782786487664010819762900415, −3.73831872696498300238540202981, −1.69717993204838124963178305823,
1.69717993204838124963178305823, 3.73831872696498300238540202981, 5.31782786487664010819762900415, 6.61245659144104597163627425129, 8.174556655790753703874742506761, 9.733394579255475606520879201282, 11.09064865224903604686586560012, 12.14473325487750635672888951581, 13.28979651028398063143819666796, 14.44794685247518163269796332226
