| L(s) = 1 | − 2.56·2-s + 3-s + 4.56·4-s − 2·5-s − 2.56·6-s − 6.56·8-s + 9-s + 5.12·10-s + 2.43·11-s + 4.56·12-s − 5.12·13-s − 2·15-s + 7.68·16-s − 0.438·17-s − 2.56·18-s − 7.12·19-s − 9.12·20-s − 6.24·22-s − 6.56·24-s − 25-s + 13.1·26-s + 27-s − 6.68·29-s + 5.12·30-s − 9.56·31-s − 6.56·32-s + 2.43·33-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.577·3-s + 2.28·4-s − 0.894·5-s − 1.04·6-s − 2.31·8-s + 0.333·9-s + 1.62·10-s + 0.735·11-s + 1.31·12-s − 1.42·13-s − 0.516·15-s + 1.92·16-s − 0.106·17-s − 0.603·18-s − 1.63·19-s − 2.03·20-s − 1.33·22-s − 1.33·24-s − 0.200·25-s + 2.57·26-s + 0.192·27-s − 1.24·29-s + 0.935·30-s − 1.71·31-s − 1.15·32-s + 0.424·33-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
| \(L(1)\) |
\(\approx\) |
\(0.3793528636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3793528636\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.56T + 2T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 + 0.438T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 9.56T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 4.43T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 5.80T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−8.169639455523469205995892284691, −7.54082604004071313771553880702, −7.07156664385614049878229263333, −6.49268627126598742717016104058, −5.31470508436499007594936147498, −4.14639898833384675754932828630, −3.50064004839999450676931208595, −2.27397502199768002540044468174, −1.85380477883707391145628812344, −0.39263781656245326990218885502,
0.39263781656245326990218885502, 1.85380477883707391145628812344, 2.27397502199768002540044468174, 3.50064004839999450676931208595, 4.14639898833384675754932828630, 5.31470508436499007594936147498, 6.49268627126598742717016104058, 7.07156664385614049878229263333, 7.54082604004071313771553880702, 8.169639455523469205995892284691
