| L(s) = 1 | − 4.49·2-s + 3·3-s + 12.2·4-s − 9.43·5-s − 13.4·6-s − 19.1·8-s + 9·9-s + 42.4·10-s − 11·11-s + 36.7·12-s − 37.2·13-s − 28.2·15-s − 12.0·16-s + 113.·17-s − 40.4·18-s − 26.0·19-s − 115.·20-s + 49.4·22-s − 66.9·23-s − 57.3·24-s − 36.0·25-s + 167.·26-s + 27·27-s − 100.·29-s + 127.·30-s + 113.·31-s + 206.·32-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 0.577·3-s + 1.53·4-s − 0.843·5-s − 0.918·6-s − 0.844·8-s + 0.333·9-s + 1.34·10-s − 0.301·11-s + 0.883·12-s − 0.795·13-s − 0.487·15-s − 0.187·16-s + 1.62·17-s − 0.530·18-s − 0.314·19-s − 1.29·20-s + 0.479·22-s − 0.606·23-s − 0.487·24-s − 0.288·25-s + 1.26·26-s + 0.192·27-s − 0.645·29-s + 0.774·30-s + 0.658·31-s + 1.14·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6630556714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6630556714\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.49T + 8T^{2} \) |
| 5 | \( 1 + 9.43T + 125T^{2} \) |
| 13 | \( 1 + 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 113.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 131.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 60.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 494.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 91.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 74.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 947.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 839.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 284.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 634.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.60e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 643.T + 9.12e5T^{2} \) |
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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
−8.951718732770354762434223351914, −8.166231935345361459231827267757, −7.70844268350432222280389810386, −7.24658179639867425577538876619, −6.09106933235815122010001669154, −4.81962730802555643560764393009, −3.72789882267103415693568210413, −2.71120332379455620294016752427, −1.65771539998880962007722702361, −0.48695820440038006845134912861,
0.48695820440038006845134912861, 1.65771539998880962007722702361, 2.71120332379455620294016752427, 3.72789882267103415693568210413, 4.81962730802555643560764393009, 6.09106933235815122010001669154, 7.24658179639867425577538876619, 7.70844268350432222280389810386, 8.166231935345361459231827267757, 8.951718732770354762434223351914
