| L(s) = 1 | + 5.99·5-s + 15.5·7-s − 34.8·11-s − 80.6·13-s + 70.1·17-s − 4.25·19-s − 118.·23-s − 89.0·25-s − 123.·29-s − 185.·31-s + 93.3·35-s + 151.·37-s − 212.·41-s + 290.·43-s + 212.·47-s − 100.·49-s + 556.·53-s − 209.·55-s + 853.·59-s − 688.·61-s − 483.·65-s − 915.·67-s − 786.·71-s − 993.·73-s − 543.·77-s − 568.·79-s + 747.·83-s + ⋯ |
| L(s) = 1 | + 0.536·5-s + 0.841·7-s − 0.956·11-s − 1.72·13-s + 1.00·17-s − 0.0513·19-s − 1.07·23-s − 0.712·25-s − 0.790·29-s − 1.07·31-s + 0.450·35-s + 0.673·37-s − 0.809·41-s + 1.02·43-s + 0.659·47-s − 0.292·49-s + 1.44·53-s − 0.512·55-s + 1.88·59-s − 1.44·61-s − 0.922·65-s − 1.66·67-s − 1.31·71-s − 1.59·73-s − 0.804·77-s − 0.809·79-s + 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 5.99T + 125T^{2} \) |
| 7 | \( 1 - 15.5T + 343T^{2} \) |
| 11 | \( 1 + 34.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 80.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 70.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4.25T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 151.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 212.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 290.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 556.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 853.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 688.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 915.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 786.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 993.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 568.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 747.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−9.949103162876949355056094231207, −8.898350995223479621495756044057, −7.66877804297434118765627876147, −7.46124561627944787166588387352, −5.80096249330672866898354371667, −5.28197025791286156375990825500, −4.19945171521150039823917148651, −2.67308267044506813294836534845, −1.76687586863298064772537913840, 0,
1.76687586863298064772537913840, 2.67308267044506813294836534845, 4.19945171521150039823917148651, 5.28197025791286156375990825500, 5.80096249330672866898354371667, 7.46124561627944787166588387352, 7.66877804297434118765627876147, 8.898350995223479621495756044057, 9.949103162876949355056094231207
