| L(s) = 1 | + 0.248·2-s − 3·3-s − 7.93·4-s − 12.4·5-s − 0.744·6-s − 3.95·8-s + 9·9-s − 3.08·10-s + 60.3·11-s + 23.8·12-s + 36.4·13-s + 37.3·15-s + 62.5·16-s − 48.7·17-s + 2.23·18-s − 50.5·19-s + 98.7·20-s + 14.9·22-s + 138.·23-s + 11.8·24-s + 29.6·25-s + 9.03·26-s − 27·27-s − 61.1·29-s + 9.25·30-s − 1.16·31-s + 47.1·32-s + ⋯ |
| L(s) = 1 | + 0.0877·2-s − 0.577·3-s − 0.992·4-s − 1.11·5-s − 0.0506·6-s − 0.174·8-s + 0.333·9-s − 0.0975·10-s + 1.65·11-s + 0.572·12-s + 0.777·13-s + 0.642·15-s + 0.976·16-s − 0.695·17-s + 0.0292·18-s − 0.610·19-s + 1.10·20-s + 0.144·22-s + 1.25·23-s + 0.100·24-s + 0.236·25-s + 0.0681·26-s − 0.192·27-s − 0.391·29-s + 0.0563·30-s − 0.00677·31-s + 0.260·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9119739018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9119739018\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.248T + 8T^{2} \) |
| 5 | \( 1 + 12.4T + 125T^{2} \) |
| 11 | \( 1 - 60.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 61.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 1.16T + 2.97e4T^{2} \) |
| 37 | \( 1 - 69.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 308.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 174.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 389.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 314.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 844.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 338.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 971.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 98.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 710.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 486.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 605.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 218.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 782.T + 9.12e5T^{2} \) |
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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
−12.51348529779422459585959971924, −11.61241142810012786116654669595, −10.81064594166399962105859860683, −9.273612063703971131222942790737, −8.596926275864234421458427459234, −7.21307866961338638517013981050, −5.98931998460235655427104357386, −4.43213723382715478558351535525, −3.79393981102392500223210340747, −0.826718000218604297585452833040,
0.826718000218604297585452833040, 3.79393981102392500223210340747, 4.43213723382715478558351535525, 5.98931998460235655427104357386, 7.21307866961338638517013981050, 8.596926275864234421458427459234, 9.273612063703971131222942790737, 10.81064594166399962105859860683, 11.61241142810012786116654669595, 12.51348529779422459585959971924
