| L(s) = 1 | + 3.87i·5-s − 3.87i·11-s + 3.46i·13-s − 4·19-s − 7.74i·23-s − 10.0·25-s + 6.70·29-s + 31-s + 4·37-s + 7.74i·41-s − 6.92i·43-s − 13.4·47-s − 6.70·53-s + 15.0·55-s − 6.70·59-s + ⋯ |
| L(s) = 1 | + 1.73i·5-s − 1.16i·11-s + 0.960i·13-s − 0.917·19-s − 1.61i·23-s − 2.00·25-s + 1.24·29-s + 0.179·31-s + 0.657·37-s + 1.20i·41-s − 1.05i·43-s − 1.95·47-s − 0.921·53-s + 2.02·55-s − 0.873·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7661369883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7661369883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.87iT - 5T^{2} \) |
| 11 | \( 1 + 3.87iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 7.74iT - 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 7.74iT - 41T^{2} \) |
| 43 | \( 1 + 6.92iT - 43T^{2} \) |
| 47 | \( 1 + 13.4T + 47T^{2} \) |
| 53 | \( 1 + 6.70T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 7.74iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 - 7.74iT - 89T^{2} \) |
| 97 | \( 1 + 5.19iT - 97T^{2} \) |
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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
−7.87624507654054761181919497559, −6.71265621882358458570433340117, −6.53479979271118022140864982210, −6.08821833581985972158921203355, −4.83086731547623608003582137870, −4.11394668414404774472396847947, −3.16843187068028648422359091220, −2.73353966047190483946519546726, −1.78078014285196046920368611178, −0.19303620294808033317587379432,
1.07619896805680273018663696559, 1.76725165559088633346176531435, 2.87842304912992421571141283935, 4.00140658454423579610073785112, 4.60420425460600731763959549275, 5.16888927474375648083528463706, 5.81211891648148629941014635827, 6.68940537866871887741477811929, 7.64069897864117434675570473412, 8.102014498580568786141979014386
