| L(s) = 1 | − 8·2-s − 448·4-s − 4.24e3·7-s + 7.68e3·8-s + 4.62e4·11-s + 1.15e5·13-s + 3.39e4·14-s + 1.67e5·16-s + 4.94e5·17-s − 1.00e6·19-s − 3.69e5·22-s − 5.32e5·23-s − 9.27e5·26-s + 1.90e6·28-s − 4.19e6·29-s − 3.36e6·31-s − 5.27e6·32-s − 3.95e6·34-s + 1.49e7·37-s + 8.06e6·38-s − 1.10e7·41-s + 6.39e6·43-s − 2.07e7·44-s + 4.26e6·46-s − 3.55e7·47-s − 2.23e7·49-s − 5.19e7·52-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s − 0.667·7-s + 0.662·8-s + 0.951·11-s + 1.12·13-s + 0.236·14-s + 0.640·16-s + 1.43·17-s − 1.77·19-s − 0.336·22-s − 0.396·23-s − 0.398·26-s + 0.584·28-s − 1.10·29-s − 0.654·31-s − 0.889·32-s − 0.508·34-s + 1.30·37-s + 0.627·38-s − 0.611·41-s + 0.285·43-s − 0.832·44-s + 0.140·46-s − 1.06·47-s − 0.554·49-s − 0.985·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.223330377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223330377\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T + p^{9} T^{2} \) |
| 7 | \( 1 + 606 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 46208 T + p^{9} T^{2} \) |
| 13 | \( 1 - 686 p^{2} T + p^{9} T^{2} \) |
| 17 | \( 1 - 494842 T + p^{9} T^{2} \) |
| 19 | \( 1 + 1008740 T + p^{9} T^{2} \) |
| 23 | \( 1 + 532554 T + p^{9} T^{2} \) |
| 29 | \( 1 + 4196390 T + p^{9} T^{2} \) |
| 31 | \( 1 + 3365028 T + p^{9} T^{2} \) |
| 37 | \( 1 - 14931358 T + p^{9} T^{2} \) |
| 41 | \( 1 + 11056262 T + p^{9} T^{2} \) |
| 43 | \( 1 - 6396794 T + p^{9} T^{2} \) |
| 47 | \( 1 + 35559158 T + p^{9} T^{2} \) |
| 53 | \( 1 - 39738586 T + p^{9} T^{2} \) |
| 59 | \( 1 - 85185620 T + p^{9} T^{2} \) |
| 61 | \( 1 - 45748642 T + p^{9} T^{2} \) |
| 67 | \( 1 - 45286158 T + p^{9} T^{2} \) |
| 71 | \( 1 - 189967468 T + p^{9} T^{2} \) |
| 73 | \( 1 + 412170946 T + p^{9} T^{2} \) |
| 79 | \( 1 - 95040840 T + p^{9} T^{2} \) |
| 83 | \( 1 - 261706326 T + p^{9} T^{2} \) |
| 89 | \( 1 - 19938630 T + p^{9} T^{2} \) |
| 97 | \( 1 - 19503358 T + p^{9} T^{2} \) |
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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
−10.38890488764788890547578760935, −9.561940847306626492392078054787, −8.762251086008796818576128750553, −7.903492376256398911775394845356, −6.54072534448370259192305111847, −5.63237986726381299992069595826, −4.15547262894331449207924142450, −3.49733041830575618968991018090, −1.66446476962926873700260963708, −0.57492763028504225207574166262,
0.57492763028504225207574166262, 1.66446476962926873700260963708, 3.49733041830575618968991018090, 4.15547262894331449207924142450, 5.63237986726381299992069595826, 6.54072534448370259192305111847, 7.903492376256398911775394845356, 8.762251086008796818576128750553, 9.561940847306626492392078054787, 10.38890488764788890547578760935
