| L(s) = 1 | + 0.642i·3-s + 3.58i·7-s + 2.58·9-s − 1.35i·13-s + 5.58i·17-s − 19-s − 2.30·21-s + 4.87i·23-s + 3.58i·27-s − 9.58·29-s − 7.17·31-s − 0.945i·37-s + 0.871·39-s + 10.4·41-s + 2.71i·43-s + ⋯ |
| L(s) = 1 | + 0.370i·3-s + 1.35i·7-s + 0.862·9-s − 0.376i·13-s + 1.35i·17-s − 0.229·19-s − 0.502·21-s + 1.01i·23-s + 0.690i·27-s − 1.78·29-s − 1.28·31-s − 0.155i·37-s + 0.139·39-s + 1.63·41-s + 0.414i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.355580032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.355580032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.642iT - 3T^{2} \) |
| 7 | \( 1 - 3.58iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 1.35iT - 13T^{2} \) |
| 17 | \( 1 - 5.58iT - 17T^{2} \) |
| 23 | \( 1 - 4.87iT - 23T^{2} \) |
| 29 | \( 1 + 9.58T + 29T^{2} \) |
| 31 | \( 1 + 7.17T + 31T^{2} \) |
| 37 | \( 1 + 0.945iT - 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 2.71iT - 43T^{2} \) |
| 47 | \( 1 + 5.89iT - 47T^{2} \) |
| 53 | \( 1 + 9.81iT - 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 3.28T + 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 4.15iT - 73T^{2} \) |
| 79 | \( 1 + 1.28T + 79T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 6.45T + 89T^{2} \) |
| 97 | \( 1 + 13.4iT - 97T^{2} \) |
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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
−8.968708136977542248444540908585, −8.086405394290603406167985462812, −7.45420776331670077984164848906, −6.51435156681031873418921353191, −5.59536003204884577479617124482, −5.33584631366324475094662907077, −4.05960019480092712013542013190, −3.56428376748819566317412804000, −2.30883090086229616369517314445, −1.56372697768125446551306778271,
0.39143926442762275536399869671, 1.41903135605133277498661569868, 2.45743926949288468244277081776, 3.69165310339962767369560678964, 4.28796108798481282090230690788, 5.02279608309189650407825623625, 6.15247918805716268122484083626, 6.87942598608891239478802721835, 7.53182315848715269852763828935, 7.74432118609443604928201055271
