| L(s) = 1 | + (1 + 1.41i)3-s − 2i·5-s + (−1.00 + 2.82i)9-s − 4.24·11-s + 1.41·13-s + (2.82 − 2i)15-s + 2i·17-s − 4.24·23-s + 25-s + (−5.00 + 1.41i)27-s + 8.48i·29-s + 8.48i·31-s + (−4.24 − 6i)33-s + 6·37-s + (1.41 + 2.00i)39-s + ⋯ |
| L(s) = 1 | + (0.577 + 0.816i)3-s − 0.894i·5-s + (−0.333 + 0.942i)9-s − 1.27·11-s + 0.392·13-s + (0.730 − 0.516i)15-s + 0.485i·17-s − 0.884·23-s + 0.200·25-s + (−0.962 + 0.272i)27-s + 1.57i·29-s + 1.52i·31-s + (−0.738 − 1.04i)33-s + 0.986·37-s + (0.226 + 0.320i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.308998288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308998288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 - 8.48iT - 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 9.89T + 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 2iT - 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−9.132475684788352914815383674983, −8.463172465702492917791477065442, −8.098161302339478705024798702755, −7.10007894484129809036975875928, −5.86263887638049320262114602028, −5.11376802846146466904406921503, −4.56711300907880807425318834722, −3.54759432855569994749618497847, −2.69893896701501539368299981896, −1.46324327517915080267546423612,
0.39297994716565131233365283295, 2.06344160931186880622935419822, 2.70020294351996517244103150118, 3.53698701559598464905588566421, 4.65766525441782818240848838128, 5.99494124088692181499266466740, 6.28529455624340743276344681500, 7.49562895629326149513664195245, 7.71869711897797973450063336712, 8.489424469925738044500604051130
