| L(s) = 1 | − 1.89·3-s + 2·5-s − 4.92·7-s + 0.601·9-s + 2.15·11-s − 3.32·13-s − 3.79·15-s + 5.91·17-s + 0.927·19-s + 9.33·21-s + 7.63·23-s − 25-s + 4.55·27-s − 1.59·29-s + 5.18·31-s − 4.09·33-s − 9.84·35-s + 1.53·37-s + 6.31·39-s − 9.71·41-s − 6.65·43-s + 1.20·45-s − 5.72·47-s + 17.2·49-s − 11.2·51-s − 9.24·53-s + 4.31·55-s + ⋯ |
| L(s) = 1 | − 1.09·3-s + 0.894·5-s − 1.85·7-s + 0.200·9-s + 0.649·11-s − 0.922·13-s − 0.979·15-s + 1.43·17-s + 0.212·19-s + 2.03·21-s + 1.59·23-s − 0.200·25-s + 0.876·27-s − 0.296·29-s + 0.930·31-s − 0.712·33-s − 1.66·35-s + 0.251·37-s + 1.01·39-s − 1.51·41-s − 1.01·43-s + 0.179·45-s − 0.834·47-s + 2.45·49-s − 1.57·51-s − 1.26·53-s + 0.581·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 1.89T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 4.92T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 + 3.32T + 13T^{2} \) |
| 17 | \( 1 - 5.91T + 17T^{2} \) |
| 19 | \( 1 - 0.927T + 19T^{2} \) |
| 23 | \( 1 - 7.63T + 23T^{2} \) |
| 29 | \( 1 + 1.59T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 - 1.53T + 37T^{2} \) |
| 41 | \( 1 + 9.71T + 41T^{2} \) |
| 43 | \( 1 + 6.65T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 + 9.24T + 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 8.23T + 79T^{2} \) |
| 83 | \( 1 + 7.97T + 83T^{2} \) |
| 89 | \( 1 + 8.41T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−8.694356781403712047634326153554, −7.37859645782419142582536951875, −6.65581436720935587331443653134, −6.20437732199346010821133203098, −5.50337042786069659679352440223, −4.83093994149858221226492397290, −3.41103333503007170916307121544, −2.82983803853728370876637143635, −1.26993248324015174257983789206, 0,
1.26993248324015174257983789206, 2.82983803853728370876637143635, 3.41103333503007170916307121544, 4.83093994149858221226492397290, 5.50337042786069659679352440223, 6.20437732199346010821133203098, 6.65581436720935587331443653134, 7.37859645782419142582536951875, 8.694356781403712047634326153554
