| L(s) = 1 | + 5·2-s − 103·4-s − 125·5-s + 930·7-s − 1.15e3·8-s − 625·10-s + 8.45e3·11-s + 6.22e3·13-s + 4.65e3·14-s + 7.40e3·16-s + 9.59e3·17-s − 4.58e4·19-s + 1.28e4·20-s + 4.22e4·22-s + 1.02e5·23-s + 1.56e4·25-s + 3.11e4·26-s − 9.57e4·28-s + 8.75e4·29-s − 7.62e4·31-s + 1.84e5·32-s + 4.79e4·34-s − 1.16e5·35-s + 2.64e5·37-s − 2.29e5·38-s + 1.44e5·40-s + 1.03e5·41-s + ⋯ |
| L(s) = 1 | + 0.441·2-s − 0.804·4-s − 0.447·5-s + 1.02·7-s − 0.797·8-s − 0.197·10-s + 1.91·11-s + 0.785·13-s + 0.452·14-s + 0.452·16-s + 0.473·17-s − 1.53·19-s + 0.359·20-s + 0.845·22-s + 1.75·23-s + 1/5·25-s + 0.347·26-s − 0.824·28-s + 0.666·29-s − 0.459·31-s + 0.997·32-s + 0.209·34-s − 0.458·35-s + 0.858·37-s − 0.678·38-s + 0.356·40-s + 0.234·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.062881210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.062881210\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p^{3} T \) |
| good | 2 | \( 1 - 5 T + p^{7} T^{2} \) |
| 7 | \( 1 - 930 T + p^{7} T^{2} \) |
| 11 | \( 1 - 8450 T + p^{7} T^{2} \) |
| 13 | \( 1 - 6220 T + p^{7} T^{2} \) |
| 17 | \( 1 - 9590 T + p^{7} T^{2} \) |
| 19 | \( 1 + 45884 T + p^{7} T^{2} \) |
| 23 | \( 1 - 4440 p T + p^{7} T^{2} \) |
| 29 | \( 1 - 87550 T + p^{7} T^{2} \) |
| 31 | \( 1 + 76212 T + p^{7} T^{2} \) |
| 37 | \( 1 - 264440 T + p^{7} T^{2} \) |
| 41 | \( 1 - 103600 T + p^{7} T^{2} \) |
| 43 | \( 1 + 324680 T + p^{7} T^{2} \) |
| 47 | \( 1 + 855880 T + p^{7} T^{2} \) |
| 53 | \( 1 - 958190 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1239550 T + p^{7} T^{2} \) |
| 61 | \( 1 - 628522 T + p^{7} T^{2} \) |
| 67 | \( 1 - 310380 T + p^{7} T^{2} \) |
| 71 | \( 1 - 3934300 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4556090 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5371644 T + p^{7} T^{2} \) |
| 83 | \( 1 - 6711060 T + p^{7} T^{2} \) |
| 89 | \( 1 - 3346500 T + p^{7} T^{2} \) |
| 97 | \( 1 - 15829730 T + p^{7} T^{2} \) |
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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
−14.55894993078942272249108685901, −13.23478361600188541960324780394, −12.02789625590058845029436131512, −11.01139014760218949717549730812, −9.167743149729741710354508711686, −8.315756541112746318253815329890, −6.46005617660228851809924050513, −4.76399221159821826867428792854, −3.71977141679191928923338646376, −1.12734747314895711452786677757,
1.12734747314895711452786677757, 3.71977141679191928923338646376, 4.76399221159821826867428792854, 6.46005617660228851809924050513, 8.315756541112746318253815329890, 9.167743149729741710354508711686, 11.01139014760218949717549730812, 12.02789625590058845029436131512, 13.23478361600188541960324780394, 14.55894993078942272249108685901
