| L(s) = 1 | − 5-s + (−2.23 + 1.41i)7-s + 1.41i·11-s − 4.57i·13-s + 4.47·17-s − 4.57i·19-s − 3.16i·23-s + 25-s + 10.5i·29-s − 1.74i·31-s + (2.23 − 1.41i)35-s + 10.4·37-s + 8.47·41-s + 8.47·43-s + 6.47·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + (−0.845 + 0.534i)7-s + 0.426i·11-s − 1.26i·13-s + 1.08·17-s − 1.04i·19-s − 0.659i·23-s + 0.200·25-s + 1.96i·29-s − 0.313i·31-s + (0.377 − 0.239i)35-s + 1.72·37-s + 1.32·41-s + 1.29·43-s + 0.944·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315389607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315389607\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2.23 - 1.41i)T \) |
| good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 4.57iT - 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 4.57iT - 19T^{2} \) |
| 23 | \( 1 + 3.16iT - 23T^{2} \) |
| 29 | \( 1 - 10.5iT - 29T^{2} \) |
| 31 | \( 1 + 1.74iT - 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 - 2.49iT - 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 9.15iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 4.91iT - 71T^{2} \) |
| 73 | \( 1 + 2.41iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 6.94T + 89T^{2} \) |
| 97 | \( 1 + 19.3iT - 97T^{2} \) |
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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
−9.512499698072577210774149164860, −8.972171450343744985532600790170, −7.893125644763738648714719875318, −7.31112504382494893054981855449, −6.25547621312898429127110977670, −5.49101894215325531795926211712, −4.50611456636985345297679076317, −3.28903739640957799550243330798, −2.63033718786387273821823845328, −0.74017769634497723793396981006,
0.991728723398788415882534976279, 2.61128972299291195155790488774, 3.79930490060728495644405467371, 4.26918156955569299867081174388, 5.79220033243376526133002388523, 6.30752486149898045194387278665, 7.49532053365361685554427908127, 7.85308881541489046907701524335, 9.125390060886890721868998983571, 9.660280526069829378369200803298
