L(s) = 1 | − 2.23·3-s + (−1.58 − 1.58i)5-s − 2.82i·7-s + 2.00·9-s + 5i·13-s + (3.53 + 3.53i)15-s + 3.16·17-s + 7.07·19-s + 6.32i·21-s + (−2.23 − 4.24i)23-s + 5.00i·25-s + 2.23·27-s + 9·29-s − 6.70i·31-s + (−4.47 + 4.47i)35-s + ⋯ |
L(s) = 1 | − 1.29·3-s + (−0.707 − 0.707i)5-s − 1.06i·7-s + 0.666·9-s + 1.38i·13-s + (0.912 + 0.912i)15-s + 0.766·17-s + 1.62·19-s + 1.38i·21-s + (−0.466 − 0.884i)23-s + 1.00i·25-s + 0.430·27-s + 1.67·29-s − 1.20i·31-s + (−0.755 + 0.755i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7507348254\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7507348254\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.58 + 1.58i)T \) |
| 23 | \( 1 + (2.23 + 4.24i)T \) |
good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 6.70iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 1.41iT - 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 - 4.47iT - 59T^{2} \) |
| 61 | \( 1 - 9.48iT - 61T^{2} \) |
| 67 | \( 1 + 2.82iT - 67T^{2} \) |
| 71 | \( 1 + 6.70iT - 71T^{2} \) |
| 73 | \( 1 + 15iT - 73T^{2} \) |
| 79 | \( 1 + 7.07T + 79T^{2} \) |
| 83 | \( 1 + 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 3.16iT - 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−9.087309720603873982473015153713, −8.064863572066996436789446266508, −7.36779367093521684911441939362, −6.61208597267571428787171838784, −5.79426534061458614067432303548, −4.71783683688826759284137417552, −4.42645583633499549209232145288, −3.28442520929020279282104616933, −1.36188594339038468361632552062, −0.45456748521317308201200426781,
1.01797766238544608743431260210, 2.84674273034664561521638067673, 3.44075438658072260046531515523, 5.00362920465238918417286151172, 5.40851430845518898505641137033, 6.17808678474965244970831874614, 6.99947044803869732449922731658, 7.890134690376555858679028414284, 8.493981556170312164528067437174, 9.830524083022657590158306749971