| L(s) = 1 | + (−1.55 − 1.60i)5-s + 1.21·7-s + 2.73·11-s − 6.99i·13-s − 0.391i·17-s − 2.94i·19-s + (−3.74 − 3.00i)23-s + (−0.138 + 4.99i)25-s − 3.30i·29-s + 7.96·31-s + (−1.89 − 1.94i)35-s + 1.84·37-s + 9.35i·41-s − 4.89·43-s − 2.87·47-s + ⋯ |
| L(s) = 1 | + (−0.697 − 0.716i)5-s + 0.459·7-s + 0.825·11-s − 1.94i·13-s − 0.0948i·17-s − 0.675i·19-s + (−0.779 − 0.625i)23-s + (−0.0277 + 0.999i)25-s − 0.613i·29-s + 1.42·31-s + (−0.320 − 0.329i)35-s + 0.303·37-s + 1.46i·41-s − 0.747·43-s − 0.419·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.319956018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319956018\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.55 + 1.60i)T \) |
| 23 | \( 1 + (3.74 + 3.00i)T \) |
| good | 7 | \( 1 - 1.21T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 + 6.99iT - 13T^{2} \) |
| 17 | \( 1 + 0.391iT - 17T^{2} \) |
| 19 | \( 1 + 2.94iT - 19T^{2} \) |
| 29 | \( 1 + 3.30iT - 29T^{2} \) |
| 31 | \( 1 - 7.96T + 31T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 - 9.35iT - 41T^{2} \) |
| 43 | \( 1 + 4.89T + 43T^{2} \) |
| 47 | \( 1 + 2.87T + 47T^{2} \) |
| 53 | \( 1 + 2.95iT - 53T^{2} \) |
| 59 | \( 1 - 8.16iT - 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 14.5iT - 71T^{2} \) |
| 73 | \( 1 - 1.70iT - 73T^{2} \) |
| 79 | \( 1 + 11.9iT - 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.237863052499491123729105037428, −7.65920958299180353750903189846, −6.65345789098435639483075619195, −5.89512810575262689117896450313, −4.96786277821830847198597975711, −4.48381477572308144872349505378, −3.53026581698276991010976917366, −2.68326914070531518135746144945, −1.30056207315335448416133966339, −0.40650694168547349543377473170,
1.41181800148399460539300893596, 2.26005762007868663389617421860, 3.51551531882080775453942931749, 4.07060113836617974364260303962, 4.74625013100439017691550597421, 5.92631519866266633890566726242, 6.69428441606988543579242184101, 7.07004402266552581418911335108, 8.042389902722446407293475950477, 8.541898129400397720416654032909
