| L(s) = 1 | + 273·3-s + 1.01e3·5-s − 3.95e3·7-s + 5.48e4·9-s + 5.09e4·11-s − 2.85e4·13-s + 2.77e5·15-s + 5.09e5·17-s + 6.26e5·19-s − 1.07e6·21-s − 6.53e5·23-s − 9.22e5·25-s + 9.59e6·27-s − 4.94e6·29-s − 4.07e6·31-s + 1.39e7·33-s − 4.01e6·35-s + 2.34e6·37-s − 7.79e6·39-s − 1.33e7·41-s + 7.83e6·43-s + 5.56e7·45-s + 3.96e7·47-s − 2.47e7·49-s + 1.39e8·51-s + 7.32e7·53-s + 5.17e7·55-s + ⋯ |
| L(s) = 1 | + 1.94·3-s + 0.726·5-s − 0.622·7-s + 2.78·9-s + 1.05·11-s − 0.277·13-s + 1.41·15-s + 1.48·17-s + 1.10·19-s − 1.21·21-s − 0.486·23-s − 0.472·25-s + 3.47·27-s − 1.29·29-s − 0.791·31-s + 2.04·33-s − 0.452·35-s + 0.206·37-s − 0.539·39-s − 0.737·41-s + 0.349·43-s + 2.02·45-s + 1.18·47-s − 0.612·49-s + 2.88·51-s + 1.27·53-s + 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(6.172217825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.172217825\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + p^{4} T \) |
| good | 3 | \( 1 - 91 p T + p^{9} T^{2} \) |
| 5 | \( 1 - 203 p T + p^{9} T^{2} \) |
| 7 | \( 1 + 565 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 50998 T + p^{9} T^{2} \) |
| 17 | \( 1 - 509757 T + p^{9} T^{2} \) |
| 19 | \( 1 - 626574 T + p^{9} T^{2} \) |
| 23 | \( 1 + 653524 T + p^{9} T^{2} \) |
| 29 | \( 1 + 4943006 T + p^{9} T^{2} \) |
| 31 | \( 1 + 4071700 T + p^{9} T^{2} \) |
| 37 | \( 1 - 2348883 T + p^{9} T^{2} \) |
| 41 | \( 1 + 13350960 T + p^{9} T^{2} \) |
| 43 | \( 1 - 7834847 T + p^{9} T^{2} \) |
| 47 | \( 1 - 39637681 T + p^{9} T^{2} \) |
| 53 | \( 1 - 73200924 T + p^{9} T^{2} \) |
| 59 | \( 1 - 141141614 T + p^{9} T^{2} \) |
| 61 | \( 1 + 132061256 T + p^{9} T^{2} \) |
| 67 | \( 1 - 185673110 T + p^{9} T^{2} \) |
| 71 | \( 1 + 224452625 T + p^{9} T^{2} \) |
| 73 | \( 1 + 2363338 p T + p^{9} T^{2} \) |
| 79 | \( 1 - 643288156 T + p^{9} T^{2} \) |
| 83 | \( 1 + 720077280 T + p^{9} T^{2} \) |
| 89 | \( 1 + 73028106 T + p^{9} T^{2} \) |
| 97 | \( 1 + 15879778 T + p^{9} T^{2} \) |
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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
−10.11556112534563397192721053078, −9.595724581033824066988389715925, −9.004143648629742625470849020360, −7.79167619409787277502681973804, −7.04453150797097506971100252588, −5.64436082348565056596482386290, −3.91746553735217312637355169837, −3.26384775424946162139280200641, −2.11826677133408916463588941064, −1.20245072118698497365043266763,
1.20245072118698497365043266763, 2.11826677133408916463588941064, 3.26384775424946162139280200641, 3.91746553735217312637355169837, 5.64436082348565056596482386290, 7.04453150797097506971100252588, 7.79167619409787277502681973804, 9.004143648629742625470849020360, 9.595724581033824066988389715925, 10.11556112534563397192721053078
