| L(s) = 1 | − 1.94·2-s + 3-s + 1.77·4-s + 2.73·5-s − 1.94·6-s + 0.439·8-s + 9-s − 5.31·10-s − 4.46·11-s + 1.77·12-s + 6.87·13-s + 2.73·15-s − 4.40·16-s + 4.96·17-s − 1.94·18-s − 1.89·19-s + 4.85·20-s + 8.68·22-s + 8.36·23-s + 0.439·24-s + 2.49·25-s − 13.3·26-s + 27-s + 5.40·29-s − 5.31·30-s + 9.94·31-s + 7.67·32-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.577·3-s + 0.886·4-s + 1.22·5-s − 0.793·6-s + 0.155·8-s + 0.333·9-s − 1.68·10-s − 1.34·11-s + 0.512·12-s + 1.90·13-s + 0.707·15-s − 1.10·16-s + 1.20·17-s − 0.457·18-s − 0.435·19-s + 1.08·20-s + 1.85·22-s + 1.74·23-s + 0.0897·24-s + 0.499·25-s − 2.61·26-s + 0.192·27-s + 1.00·29-s − 0.971·30-s + 1.78·31-s + 1.35·32-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
| \(L(1)\) |
\(\approx\) |
\(1.832163819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.832163819\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.94T + 2T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 + 4.46T + 11T^{2} \) |
| 13 | \( 1 - 6.87T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 + 1.89T + 19T^{2} \) |
| 23 | \( 1 - 8.36T + 23T^{2} \) |
| 29 | \( 1 - 5.40T + 29T^{2} \) |
| 31 | \( 1 - 9.94T + 31T^{2} \) |
| 37 | \( 1 - 3.92T + 37T^{2} \) |
| 43 | \( 1 + 0.181T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 0.454T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 - 1.59T + 61T^{2} \) |
| 67 | \( 1 + 8.14T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 7.69T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 - 5.84T + 83T^{2} \) |
| 89 | \( 1 - 9.20T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.259496606631467325424036253015, −7.76619073169320407809793436642, −6.73891265526754567163747979761, −6.20852030512435611582297769031, −5.31721409329692048063351312705, −4.53389503160756056450410475954, −3.19359308189979672091746012673, −2.61312769806565579946685150962, −1.52884156475195510082116115697, −0.957787657259516074307801453580,
0.957787657259516074307801453580, 1.52884156475195510082116115697, 2.61312769806565579946685150962, 3.19359308189979672091746012673, 4.53389503160756056450410475954, 5.31721409329692048063351312705, 6.20852030512435611582297769031, 6.73891265526754567163747979761, 7.76619073169320407809793436642, 8.259496606631467325424036253015
