L(s) = 1 | − 3-s + 4i·5-s − 2i·7-s + 9-s − 4i·11-s − 4i·15-s − 2·17-s + 2i·19-s + 2i·21-s − 11·25-s − 27-s − 6·29-s + 10i·31-s + 4i·33-s + 8·35-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.78i·5-s − 0.755i·7-s + 0.333·9-s − 1.20i·11-s − 1.03i·15-s − 0.485·17-s + 0.458i·19-s + 0.436i·21-s − 2.20·25-s − 0.192·27-s − 1.11·29-s + 1.79i·31-s + 0.696i·33-s + 1.35·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 4iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 16iT - 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 4iT - 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.813732844963216489184794391429, −7.80433978468021520564411993845, −7.12695148958217466605537996887, −6.47007881210334126460844651450, −5.92532997716241188678680318446, −4.79193386425839096467955534518, −3.54236846157479395782221387964, −3.17816743860810052265591214063, −1.72379456102557725178058192806, 0,
1.42242276437796690933912837265, 2.35866999897939738587899993063, 4.17737100000295686817537665077, 4.58728859259556129333279487065, 5.48453599620452708294071401486, 5.98465857057036710103781826432, 7.21570051869570027856824075318, 7.935730992416060941358040400705, 8.868846233846508506388452521795