| L(s) = 1 | − 1.21·2-s + 3-s − 0.534·4-s − 0.826·5-s − 1.21·6-s + 3.06·8-s + 9-s + 1.00·10-s − 0.396·11-s − 0.534·12-s − 3.17·13-s − 0.826·15-s − 2.64·16-s − 0.159·17-s − 1.21·18-s − 3.91·19-s + 0.441·20-s + 0.480·22-s + 7.13·23-s + 3.06·24-s − 4.31·25-s + 3.84·26-s + 27-s + 9.77·29-s + 1.00·30-s − 6.15·31-s − 2.93·32-s + ⋯ |
| L(s) = 1 | − 0.856·2-s + 0.577·3-s − 0.267·4-s − 0.369·5-s − 0.494·6-s + 1.08·8-s + 0.333·9-s + 0.316·10-s − 0.119·11-s − 0.154·12-s − 0.879·13-s − 0.213·15-s − 0.661·16-s − 0.0385·17-s − 0.285·18-s − 0.898·19-s + 0.0988·20-s + 0.102·22-s + 1.48·23-s + 0.626·24-s − 0.863·25-s + 0.753·26-s + 0.192·27-s + 1.81·29-s + 0.182·30-s − 1.10·31-s − 0.518·32-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 5 | \( 1 + 0.826T + 5T^{2} \) |
| 11 | \( 1 + 0.396T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 0.159T + 17T^{2} \) |
| 19 | \( 1 + 3.91T + 19T^{2} \) |
| 23 | \( 1 - 7.13T + 23T^{2} \) |
| 29 | \( 1 - 9.77T + 29T^{2} \) |
| 31 | \( 1 + 6.15T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 43 | \( 1 + 5.32T + 43T^{2} \) |
| 47 | \( 1 - 1.31T + 47T^{2} \) |
| 53 | \( 1 + 5.54T + 53T^{2} \) |
| 59 | \( 1 + 1.57T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 1.25T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 + 9.77T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.943551807567387734579502459503, −7.22469376240437062479005549181, −6.68082106537282012700934527594, −5.48019750133830327454653025699, −4.63103799388543522153571827264, −4.16485209712632947527520029645, −3.08127075424410409927863243272, −2.24337426657071445423425871729, −1.15705411750442199800909476768, 0,
1.15705411750442199800909476768, 2.24337426657071445423425871729, 3.08127075424410409927863243272, 4.16485209712632947527520029645, 4.63103799388543522153571827264, 5.48019750133830327454653025699, 6.68082106537282012700934527594, 7.22469376240437062479005549181, 7.943551807567387734579502459503
