| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 11-s − 12-s + 13-s + 16-s − 17-s + 18-s + 19-s − 22-s − 23-s − 24-s + 26-s − 27-s + 31-s + 32-s + 33-s − 34-s + 36-s + 37-s + 38-s − 39-s + 41-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 11-s − 12-s + 13-s + 16-s − 17-s + 18-s + 19-s − 22-s − 23-s − 24-s + 26-s − 27-s + 31-s + 32-s + 33-s − 34-s + 36-s + 37-s + 38-s − 39-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.177417866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.177417866\) |
| \(L(1)\) |
\(\approx\) |
\(1.577745078\) |
| \(L(1)\) |
\(\approx\) |
\(1.577745078\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.56355265184839294876959563862, −20.86679515956299500217131605120, −20.17280667317797177018595527486, −19.07778736919058278519396520070, −18.11739175841773027331019955503, −17.57988659319519482803855748566, −16.31096815350914565133093057002, −15.93470188710779499920435412319, −15.36430633199742557021518819784, −14.11847923841226337537700916379, −13.332258843873142812138988536346, −12.7899708253509177741247006216, −11.83261995570218401261630431182, −11.17834816403664234795066376849, −10.56147025019480424459085091823, −9.632860954666811255553968834721, −8.1114079582560447250033138952, −7.340881658402777042634595905495, −6.264923525379399743987209928497, −5.8457799260527501536976902892, −4.83049946284391181467857910586, −4.18704762425978835456597748613, −3.0397665227656453764020113985, −1.917548797735208455467412735501, −0.75311925095360999374078083159,
0.75311925095360999374078083159, 1.917548797735208455467412735501, 3.0397665227656453764020113985, 4.18704762425978835456597748613, 4.83049946284391181467857910586, 5.8457799260527501536976902892, 6.264923525379399743987209928497, 7.340881658402777042634595905495, 8.1114079582560447250033138952, 9.632860954666811255553968834721, 10.56147025019480424459085091823, 11.17834816403664234795066376849, 11.83261995570218401261630431182, 12.7899708253509177741247006216, 13.332258843873142812138988536346, 14.11847923841226337537700916379, 15.36430633199742557021518819784, 15.93470188710779499920435412319, 16.31096815350914565133093057002, 17.57988659319519482803855748566, 18.11739175841773027331019955503, 19.07778736919058278519396520070, 20.17280667317797177018595527486, 20.86679515956299500217131605120, 21.56355265184839294876959563862
