| L(s) = 1 | − 3.99·2-s + 7.97·4-s − 4.04·5-s − 1.10·7-s + 0.0929·8-s + 16.1·10-s + 47.9·11-s + 50.4·13-s + 4.41·14-s − 64.1·16-s − 51.1·17-s − 87.9·19-s − 32.2·20-s − 191.·22-s − 141.·23-s − 108.·25-s − 201.·26-s − 8.81·28-s + 88.7·29-s + 242.·31-s + 255.·32-s + 204.·34-s + 4.47·35-s + 93.5·37-s + 351.·38-s − 0.376·40-s + 437.·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.997·4-s − 0.362·5-s − 0.0596·7-s + 0.00410·8-s + 0.511·10-s + 1.31·11-s + 1.07·13-s + 0.0842·14-s − 1.00·16-s − 0.729·17-s − 1.06·19-s − 0.361·20-s − 1.85·22-s − 1.28·23-s − 0.868·25-s − 1.52·26-s − 0.0594·28-s + 0.568·29-s + 1.40·31-s + 1.41·32-s + 1.03·34-s + 0.0215·35-s + 0.415·37-s + 1.50·38-s − 0.00148·40-s + 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8854363348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8854363348\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 3.99T + 8T^{2} \) |
| 5 | \( 1 + 4.04T + 125T^{2} \) |
| 7 | \( 1 + 1.10T + 343T^{2} \) |
| 11 | \( 1 - 47.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 50.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 87.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 141.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 88.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 93.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 437.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 173.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 118.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 674.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 122.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 711.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.22e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 13.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 113.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 476.T + 9.12e5T^{2} \) |
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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
−8.590276642495815905185994713197, −8.304895831631034619496492328764, −7.39551813143789931453856780335, −6.46528849108475277126318898331, −6.06857847112150271669853039586, −4.30384754756943944386803909100, −4.03080302133824285322101343203, −2.46223587332802827375039833436, −1.47818612280768709096156275219, −0.56783342640752875031269933021,
0.56783342640752875031269933021, 1.47818612280768709096156275219, 2.46223587332802827375039833436, 4.03080302133824285322101343203, 4.30384754756943944386803909100, 6.06857847112150271669853039586, 6.46528849108475277126318898331, 7.39551813143789931453856780335, 8.304895831631034619496492328764, 8.590276642495815905185994713197
