| L(s) = 1 | − 6.11·3-s − 11.9·5-s − 7·7-s + 10.3·9-s + 38.7·11-s + 90.3·13-s + 72.8·15-s − 17·17-s + 46.0·19-s + 42.7·21-s − 76.9·23-s + 16.9·25-s + 101.·27-s + 5.61·29-s − 193.·31-s − 236.·33-s + 83.4·35-s − 280.·37-s − 552.·39-s + 272.·41-s − 374.·43-s − 123.·45-s + 458.·47-s + 49·49-s + 103.·51-s + 272.·53-s − 461.·55-s + ⋯ |
| L(s) = 1 | − 1.17·3-s − 1.06·5-s − 0.377·7-s + 0.382·9-s + 1.06·11-s + 1.92·13-s + 1.25·15-s − 0.242·17-s + 0.555·19-s + 0.444·21-s − 0.697·23-s + 0.135·25-s + 0.725·27-s + 0.0359·29-s − 1.12·31-s − 1.24·33-s + 0.402·35-s − 1.24·37-s − 2.26·39-s + 1.03·41-s − 1.32·43-s − 0.407·45-s + 1.42·47-s + 0.142·49-s + 0.285·51-s + 0.707·53-s − 1.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 6.11T + 27T^{2} \) |
| 5 | \( 1 + 11.9T + 125T^{2} \) |
| 11 | \( 1 - 38.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 90.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 46.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 5.61T + 2.43e4T^{2} \) |
| 31 | \( 1 + 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 280.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 272.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 374.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 458.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 272.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 462.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 300.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 216.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 10.9T + 3.57e5T^{2} \) |
| 73 | \( 1 - 281.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 634.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 625.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 584.T + 9.12e5T^{2} \) |
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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
−10.50947901467787511692015957966, −9.165191617795516877488095866999, −8.383147028126122527422503097233, −7.18489523427932593522882176754, −6.31229565558464711513880030156, −5.59161667534341903528611464291, −4.17086979251725682066408233631, −3.50363061292864322173815245500, −1.25429465212708516652134763763, 0,
1.25429465212708516652134763763, 3.50363061292864322173815245500, 4.17086979251725682066408233631, 5.59161667534341903528611464291, 6.31229565558464711513880030156, 7.18489523427932593522882176754, 8.383147028126122527422503097233, 9.165191617795516877488095866999, 10.50947901467787511692015957966
