| L(s) = 1 | + 88.9·3-s − 1.18e3·5-s − 3.81e3·7-s − 1.17e4·9-s + 4.45e4·11-s + 2.85e4·13-s − 1.05e5·15-s − 3.33e5·17-s − 6.29e4·19-s − 3.39e5·21-s + 2.12e6·23-s − 5.40e5·25-s − 2.79e6·27-s − 7.05e6·29-s − 8.97e6·31-s + 3.96e6·33-s + 4.53e6·35-s + 1.61e7·37-s + 2.54e6·39-s + 3.64e6·41-s + 3.21e7·43-s + 1.39e7·45-s − 1.39e7·47-s − 2.57e7·49-s − 2.96e7·51-s + 7.19e7·53-s − 5.28e7·55-s + ⋯ |
| L(s) = 1 | + 0.634·3-s − 0.850·5-s − 0.601·7-s − 0.597·9-s + 0.916·11-s + 0.277·13-s − 0.539·15-s − 0.967·17-s − 0.110·19-s − 0.381·21-s + 1.58·23-s − 0.276·25-s − 1.01·27-s − 1.85·29-s − 1.74·31-s + 0.581·33-s + 0.511·35-s + 1.41·37-s + 0.175·39-s + 0.201·41-s + 1.43·43-s + 0.508·45-s − 0.416·47-s − 0.638·49-s − 0.613·51-s + 1.25·53-s − 0.779·55-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
| \(L(5)\) |
\(\approx\) |
\(1.579059805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.579059805\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 - 88.9T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.18e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.81e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.45e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.33e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.29e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.05e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.61e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.64e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.21e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.39e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.52e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.09e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.66e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.43e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.49e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.57e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.21e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.63e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.61e8T + 7.60e17T^{2} \) |
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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
−11.05856511875742163700702938818, −9.315807231752536827047240023686, −8.978384477381700088048619652038, −7.77445711179147372918380448480, −6.86369616288356250978976028990, −5.65980851715239396765194152102, −4.07291288932415465186173988847, −3.40798115652512684159876386859, −2.15698454019157483283470646731, −0.56018285910349227907447937063,
0.56018285910349227907447937063, 2.15698454019157483283470646731, 3.40798115652512684159876386859, 4.07291288932415465186173988847, 5.65980851715239396765194152102, 6.86369616288356250978976028990, 7.77445711179147372918380448480, 8.978384477381700088048619652038, 9.315807231752536827047240023686, 11.05856511875742163700702938818
