| L(s) = 1 | + 4.47·2-s + 12.0·4-s − 13.5·5-s − 3.12·7-s + 18.1·8-s − 60.8·10-s + 36.2·11-s − 24.2·13-s − 13.9·14-s − 15.2·16-s + 46.4·17-s + 69.2·19-s − 163.·20-s + 162.·22-s + 58.5·23-s + 59.7·25-s − 108.·26-s − 37.5·28-s − 32.3·29-s − 37.0·31-s − 213.·32-s + 207.·34-s + 42.4·35-s − 48.5·37-s + 309.·38-s − 246.·40-s + 37.0·41-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.50·4-s − 1.21·5-s − 0.168·7-s + 0.800·8-s − 1.92·10-s + 0.992·11-s − 0.517·13-s − 0.266·14-s − 0.238·16-s + 0.662·17-s + 0.835·19-s − 1.83·20-s + 1.57·22-s + 0.530·23-s + 0.477·25-s − 0.818·26-s − 0.253·28-s − 0.207·29-s − 0.214·31-s − 1.17·32-s + 1.04·34-s + 0.204·35-s − 0.215·37-s + 1.32·38-s − 0.972·40-s + 0.141·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 4.47T + 8T^{2} \) |
| 5 | \( 1 + 13.5T + 125T^{2} \) |
| 7 | \( 1 + 3.12T + 343T^{2} \) |
| 11 | \( 1 - 36.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 24.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 58.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 32.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 37.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 48.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 37.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 274.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 17.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 670.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 26.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 175.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 200.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 616.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 855.T + 9.12e5T^{2} \) |
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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
−8.081102822049180827684888286790, −7.28119205943749455632167521464, −6.71382197067409914684774486741, −5.77030631767954994980073362516, −4.94842304050689672841973642438, −4.23630162496174389055516512996, −3.48940199746572661085539352841, −2.95786907652220002769866673956, −1.47638507357098163945627320622, 0,
1.47638507357098163945627320622, 2.95786907652220002769866673956, 3.48940199746572661085539352841, 4.23630162496174389055516512996, 4.94842304050689672841973642438, 5.77030631767954994980073362516, 6.71382197067409914684774486741, 7.28119205943749455632167521464, 8.081102822049180827684888286790
