| L(s) = 1 | + (−1.03 − 1.38i)3-s + (−1.36 − 1.77i)5-s + (−0.844 + 2.87i)9-s − 4.03i·11-s − 4.67·13-s + (−1.04 + 3.73i)15-s + 6.20i·17-s − 7.23i·19-s − 2.35·23-s + (−1.28 + 4.83i)25-s + (4.86 − 1.81i)27-s + 5.15i·29-s − 9.60i·31-s + (−5.59 + 4.19i)33-s − 5.26i·37-s + ⋯ |
| L(s) = 1 | + (−0.599 − 0.800i)3-s + (−0.609 − 0.792i)5-s + (−0.281 + 0.959i)9-s − 1.21i·11-s − 1.29·13-s + (−0.268 + 0.963i)15-s + 1.50i·17-s − 1.65i·19-s − 0.490·23-s + (−0.256 + 0.966i)25-s + (0.936 − 0.349i)27-s + 0.957i·29-s − 1.72i·31-s + (−0.974 + 0.729i)33-s − 0.866i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2731912595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2731912595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.03 + 1.38i)T \) |
| 5 | \( 1 + (1.36 + 1.77i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4.03iT - 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 - 6.20iT - 17T^{2} \) |
| 19 | \( 1 + 7.23iT - 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 - 5.15iT - 29T^{2} \) |
| 31 | \( 1 + 9.60iT - 31T^{2} \) |
| 37 | \( 1 + 5.26iT - 37T^{2} \) |
| 41 | \( 1 + 9.85T + 41T^{2} \) |
| 43 | \( 1 - 12.7iT - 43T^{2} \) |
| 47 | \( 1 - 7.80iT - 47T^{2} \) |
| 53 | \( 1 - 2.63T + 53T^{2} \) |
| 59 | \( 1 + 4.28T + 59T^{2} \) |
| 61 | \( 1 + 1.43iT - 61T^{2} \) |
| 67 | \( 1 + 3.05iT - 67T^{2} \) |
| 71 | \( 1 - 6.25iT - 71T^{2} \) |
| 73 | \( 1 + 2.25T + 73T^{2} \) |
| 79 | \( 1 - 7.22T + 79T^{2} \) |
| 83 | \( 1 + 4.50iT - 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 5.21T + 97T^{2} \) |
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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
−8.695895103609608681989509150397, −7.979699971327284158771514068540, −7.50019948227055624308819093192, −6.53246138208049954974280414368, −5.84849496905838604434279234219, −5.02936775074975404853930404006, −4.35492589438671590863577038045, −3.16991515402375006638877831734, −2.04897449719455172140692894651, −0.846842535532453819619346483497,
0.12194196508611776019783514565, 2.06254170839862002466199649102, 3.14766806544057666310084081616, 3.94033113098655909966507498909, 4.81669895078440929985116026730, 5.32808922744047502242744580680, 6.51620127696711081989828446759, 7.08519884112134650125977329404, 7.71460611199176015785209887125, 8.714632612868741743540135285102
