| L(s) = 1 | + i·2-s + 2.24·3-s − 4-s + 1.69i·5-s + 2.24i·6-s + i·7-s − i·8-s + 2.04·9-s − 1.69·10-s − 0.445i·11-s − 2.24·12-s − 14-s + 3.80i·15-s + 16-s + 2.15·17-s + 2.04i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.29·3-s − 0.5·4-s + 0.756i·5-s + 0.917i·6-s + 0.377i·7-s − 0.353i·8-s + 0.682·9-s − 0.535·10-s − 0.134i·11-s − 0.648·12-s − 0.267·14-s + 0.981i·15-s + 0.250·16-s + 0.523·17-s + 0.482i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.471960069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.471960069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 5 | \( 1 - 1.69iT - 5T^{2} \) |
| 11 | \( 1 + 0.445iT - 11T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 - 6.35iT - 19T^{2} \) |
| 23 | \( 1 - 0.911T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 - 8.89iT - 31T^{2} \) |
| 37 | \( 1 + 10.8iT - 37T^{2} \) |
| 41 | \( 1 - 2.41iT - 41T^{2} \) |
| 43 | \( 1 + 4.63T + 43T^{2} \) |
| 47 | \( 1 - 9.75iT - 47T^{2} \) |
| 53 | \( 1 + 8.74T + 53T^{2} \) |
| 59 | \( 1 - 10.1iT - 59T^{2} \) |
| 61 | \( 1 - 1.37T + 61T^{2} \) |
| 67 | \( 1 - 6.23iT - 67T^{2} \) |
| 71 | \( 1 + 5.76iT - 71T^{2} \) |
| 73 | \( 1 - 9.93iT - 73T^{2} \) |
| 79 | \( 1 + 6.30T + 79T^{2} \) |
| 83 | \( 1 + 2.91iT - 83T^{2} \) |
| 89 | \( 1 + 18.1iT - 89T^{2} \) |
| 97 | \( 1 + 3.30iT - 97T^{2} \) |
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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
−8.945525494255832460957297605797, −8.500294194011572659625092337291, −7.73500279065405592550949002656, −7.17347322609816582729843621118, −6.23502697115168465639902793913, −5.51770112676679237543130704812, −4.36214372673147758244424923085, −3.34683754475820284407068855787, −2.87859949027161696623324071178, −1.59774221681403610716935203947,
0.71505184095648444841095940998, 1.90319619630997991075148312164, 2.84828093736530380785753188151, 3.55991163832477914358901714276, 4.55352126356829869017900139033, 5.12874835855494480048823874723, 6.48052152903072184467305913912, 7.44468961662395677106262402598, 8.284261481958213881878101299304, 8.641126620715576961346689289877
