| L(s) = 1 | + 5-s − 0.826i·7-s + 5.34·11-s − 4.20·13-s − 3.60·17-s − 5.22i·19-s + (4.78 − 0.321i)23-s + 25-s + 5.68i·29-s + 5.23·31-s − 0.826i·35-s + 4.95i·37-s − 11.4i·41-s − 1.85i·43-s − 1.79i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.312i·7-s + 1.61·11-s − 1.16·13-s − 0.874·17-s − 1.19i·19-s + (0.997 − 0.0669i)23-s + 0.200·25-s + 1.05i·29-s + 0.939·31-s − 0.139i·35-s + 0.814i·37-s − 1.78i·41-s − 0.283i·43-s − 0.261i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.067641827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.067641827\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + (-4.78 + 0.321i)T \) |
| good | 7 | \( 1 + 0.826iT - 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 + 4.20T + 13T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 + 5.22iT - 19T^{2} \) |
| 29 | \( 1 - 5.68iT - 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 - 4.95iT - 37T^{2} \) |
| 41 | \( 1 + 11.4iT - 41T^{2} \) |
| 43 | \( 1 + 1.85iT - 43T^{2} \) |
| 47 | \( 1 + 1.79iT - 47T^{2} \) |
| 53 | \( 1 + 6.66T + 53T^{2} \) |
| 59 | \( 1 + 4.65iT - 59T^{2} \) |
| 61 | \( 1 + 9.82iT - 61T^{2} \) |
| 67 | \( 1 - 5.81iT - 67T^{2} \) |
| 71 | \( 1 + 11.1iT - 71T^{2} \) |
| 73 | \( 1 - 6.11T + 73T^{2} \) |
| 79 | \( 1 + 1.46iT - 79T^{2} \) |
| 83 | \( 1 - 8.97T + 83T^{2} \) |
| 89 | \( 1 + 0.0563T + 89T^{2} \) |
| 97 | \( 1 - 15.0iT - 97T^{2} \) |
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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
−8.510448304212859931857235735260, −7.33457511993829169330318670320, −6.81291127556992056597323323287, −6.38135299428823504569449873317, −5.12283224038715574137479714920, −4.68918390410392736008992272813, −3.72890511402712393960323900079, −2.75491411122345292977136165013, −1.82008824296469866441640279132, −0.66232063644843579165512816963,
1.09199003691296387150410668092, 2.10533086843913290196464993777, 2.96203123353262986615478984850, 4.09770563297063129543370190693, 4.66131854054260325414732276236, 5.68252845442203554895540441508, 6.34993661961461564099958741402, 6.92572423779364531192339225201, 7.80607034344773799406425521949, 8.597986034762285460479934491377
