| L(s) = 1 | − 0.818·2-s + 3-s − 1.32·4-s + 3.72·5-s − 0.818·6-s + 2.72·8-s + 9-s − 3.04·10-s + 3.70·11-s − 1.32·12-s − 6.90·13-s + 3.72·15-s + 0.428·16-s − 3.21·17-s − 0.818·18-s + 6.07·19-s − 4.94·20-s − 3.02·22-s − 23-s + 2.72·24-s + 8.84·25-s + 5.65·26-s + 27-s − 3.33·29-s − 3.04·30-s + 0.638·31-s − 5.80·32-s + ⋯ |
| L(s) = 1 | − 0.578·2-s + 0.577·3-s − 0.664·4-s + 1.66·5-s − 0.334·6-s + 0.963·8-s + 0.333·9-s − 0.963·10-s + 1.11·11-s − 0.383·12-s − 1.91·13-s + 0.960·15-s + 0.107·16-s − 0.780·17-s − 0.192·18-s + 1.39·19-s − 1.10·20-s − 0.645·22-s − 0.208·23-s + 0.556·24-s + 1.76·25-s + 1.10·26-s + 0.192·27-s − 0.618·29-s − 0.555·30-s + 0.114·31-s − 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.072864523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.072864523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.818T + 2T^{2} \) |
| 5 | \( 1 - 3.72T + 5T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 + 6.90T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 19 | \( 1 - 6.07T + 19T^{2} \) |
| 29 | \( 1 + 3.33T + 29T^{2} \) |
| 31 | \( 1 - 0.638T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 4.34T + 43T^{2} \) |
| 47 | \( 1 + 6.02T + 47T^{2} \) |
| 53 | \( 1 - 7.27T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 + 4.01T + 67T^{2} \) |
| 71 | \( 1 + 2.92T + 71T^{2} \) |
| 73 | \( 1 - 9.14T + 73T^{2} \) |
| 79 | \( 1 - 8.17T + 79T^{2} \) |
| 83 | \( 1 - 6.01T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−9.044537559042582373020933837376, −7.899749891220175872767941663983, −7.29824709176916678486608893895, −6.46579731751539223495254830182, −5.51914198952956595250925072968, −4.83952467829695351262017731547, −4.01406488376367009788096586280, −2.70118135464532415031873530675, −1.98549291678652335157863152148, −0.960576602974353706778534829437,
0.960576602974353706778534829437, 1.98549291678652335157863152148, 2.70118135464532415031873530675, 4.01406488376367009788096586280, 4.83952467829695351262017731547, 5.51914198952956595250925072968, 6.46579731751539223495254830182, 7.29824709176916678486608893895, 7.899749891220175872767941663983, 9.044537559042582373020933837376
