| L(s) = 1 | + 5-s + 4·7-s + 11-s − 6·17-s − 4·19-s − 4·23-s + 25-s − 10·29-s + 6·31-s + 4·35-s + 2·37-s − 4·41-s + 10·43-s + 8·47-s + 9·49-s + 4·53-s + 55-s + 12·59-s − 6·61-s − 2·67-s + 8·71-s + 6·73-s + 4·77-s − 10·79-s − 8·83-s − 6·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.301·11-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.85·29-s + 1.07·31-s + 0.676·35-s + 0.328·37-s − 0.624·41-s + 1.52·43-s + 1.16·47-s + 9/7·49-s + 0.549·53-s + 0.134·55-s + 1.56·59-s − 0.768·61-s − 0.244·67-s + 0.949·71-s + 0.702·73-s + 0.455·77-s − 1.12·79-s − 0.878·83-s − 0.650·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.956006608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.956006608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.63860170077495, −11.87331408191196, −11.81478473987881, −11.10549296080870, −10.82244943312916, −10.54495536540956, −9.767971968152708, −9.417579991982736, −8.740737208477737, −8.598481514910855, −8.042856507067190, −7.519945979427032, −7.071077953690270, −6.519718782793496, −5.971127260768022, −5.591004352487923, −5.018097338444672, −4.423157895467319, −4.158570859271680, −3.679207064021969, −2.586144058513226, −2.263346498165616, −1.850409348571486, −1.223050501826780, −0.4437899161180274,
0.4437899161180274, 1.223050501826780, 1.850409348571486, 2.263346498165616, 2.586144058513226, 3.679207064021969, 4.158570859271680, 4.423157895467319, 5.018097338444672, 5.591004352487923, 5.971127260768022, 6.519718782793496, 7.071077953690270, 7.519945979427032, 8.042856507067190, 8.598481514910855, 8.740737208477737, 9.417579991982736, 9.767971968152708, 10.54495536540956, 10.82244943312916, 11.10549296080870, 11.81478473987881, 11.87331408191196, 12.63860170077495
