| L(s) = 1 | − 1.89·2-s + 3-s + 1.57·4-s + 1.57·5-s − 1.89·6-s + 0.800·8-s + 9-s − 2.98·10-s + 1.66·11-s + 1.57·12-s + 5.04·13-s + 1.57·15-s − 4.66·16-s − 6.35·17-s − 1.89·18-s − 0.691·19-s + 2.48·20-s − 3.15·22-s + 8.82·23-s + 0.800·24-s − 2.51·25-s − 9.54·26-s + 27-s − 6.64·29-s − 2.98·30-s − 2.35·31-s + 7.22·32-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.577·3-s + 0.788·4-s + 0.705·5-s − 0.772·6-s + 0.283·8-s + 0.333·9-s − 0.942·10-s + 0.502·11-s + 0.455·12-s + 1.39·13-s + 0.407·15-s − 1.16·16-s − 1.54·17-s − 0.445·18-s − 0.158·19-s + 0.555·20-s − 0.672·22-s + 1.84·23-s + 0.163·24-s − 0.502·25-s − 1.87·26-s + 0.192·27-s − 1.23·29-s − 0.544·30-s − 0.423·31-s + 1.27·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.519713738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.519713738\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 5 | \( 1 - 1.57T + 5T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 + 6.35T + 17T^{2} \) |
| 19 | \( 1 + 0.691T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 + 2.35T + 31T^{2} \) |
| 37 | \( 1 - 9.02T + 37T^{2} \) |
| 43 | \( 1 + 9.96T + 43T^{2} \) |
| 47 | \( 1 - 1.33T + 47T^{2} \) |
| 53 | \( 1 - 1.66T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 0.870T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 2.79T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 2.62T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 3.21T + 97T^{2} \) |
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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
−8.370830799494059086717251358891, −7.53705390074819106309444506936, −6.79402654142923043069727024570, −6.31953534434513971524197563743, −5.28373734483245046359074436086, −4.30784490054688919712410742117, −3.55183377681472187776939238284, −2.35233810266798006277017958435, −1.72342363493978399200393971813, −0.808217620795406463893468902869,
0.808217620795406463893468902869, 1.72342363493978399200393971813, 2.35233810266798006277017958435, 3.55183377681472187776939238284, 4.30784490054688919712410742117, 5.28373734483245046359074436086, 6.31953534434513971524197563743, 6.79402654142923043069727024570, 7.53705390074819106309444506936, 8.370830799494059086717251358891
