| L(s) = 1 | − 2-s − 2.82·3-s + 4-s − 0.629·5-s + 2.82·6-s − 0.109·7-s − 8-s + 4.98·9-s + 0.629·10-s + 5.07·11-s − 2.82·12-s − 5.87·13-s + 0.109·14-s + 1.77·15-s + 16-s − 4.42·17-s − 4.98·18-s + 6.58·19-s − 0.629·20-s + 0.309·21-s − 5.07·22-s − 3.36·23-s + 2.82·24-s − 4.60·25-s + 5.87·26-s − 5.59·27-s − 0.109·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.63·3-s + 0.5·4-s − 0.281·5-s + 1.15·6-s − 0.0413·7-s − 0.353·8-s + 1.66·9-s + 0.199·10-s + 1.52·11-s − 0.815·12-s − 1.62·13-s + 0.0292·14-s + 0.459·15-s + 0.250·16-s − 1.07·17-s − 1.17·18-s + 1.51·19-s − 0.140·20-s + 0.0674·21-s − 1.08·22-s − 0.701·23-s + 0.576·24-s − 0.920·25-s + 1.15·26-s − 1.07·27-s − 0.0206·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4840951090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4840951090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 0.629T + 5T^{2} \) |
| 7 | \( 1 + 0.109T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 13 | \( 1 + 5.87T + 13T^{2} \) |
| 17 | \( 1 + 4.42T + 17T^{2} \) |
| 19 | \( 1 - 6.58T + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 37 | \( 1 - 3.03T + 37T^{2} \) |
| 41 | \( 1 - 8.73T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 + 3.86T + 53T^{2} \) |
| 59 | \( 1 + 6.55T + 59T^{2} \) |
| 61 | \( 1 - 0.935T + 61T^{2} \) |
| 67 | \( 1 - 2.30T + 67T^{2} \) |
| 71 | \( 1 + 9.57T + 71T^{2} \) |
| 73 | \( 1 - 6.74T + 73T^{2} \) |
| 79 | \( 1 + 9.61T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 0.349T + 89T^{2} \) |
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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
−7.65732390535163906233379526515, −7.47635459594900522975680946087, −6.59015310908849419556535332572, −6.11450809901932087931497867706, −5.36170673690349649388064360041, −4.52124401471153109595266619668, −3.89880407890248221716812326673, −2.52857049183542738718941294230, −1.44659202335798026773980469591, −0.46920277421577940749920728129,
0.46920277421577940749920728129, 1.44659202335798026773980469591, 2.52857049183542738718941294230, 3.89880407890248221716812326673, 4.52124401471153109595266619668, 5.36170673690349649388064360041, 6.11450809901932087931497867706, 6.59015310908849419556535332572, 7.47635459594900522975680946087, 7.65732390535163906233379526515
