| L(s) = 1 | − 2·2-s + 4·4-s − 14.9·5-s − 21.7·7-s − 8·8-s + 29.8·10-s + 44.0·13-s + 43.5·14-s + 16·16-s − 24.9·17-s − 21.9·19-s − 59.6·20-s − 177.·23-s + 97.5·25-s − 88.0·26-s − 87.0·28-s − 149.·29-s + 75.1·31-s − 32·32-s + 49.8·34-s + 324.·35-s + 222.·37-s + 43.9·38-s + 119.·40-s − 253.·41-s − 130.·43-s + 355.·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.33·5-s − 1.17·7-s − 0.353·8-s + 0.943·10-s + 0.939·13-s + 0.831·14-s + 0.250·16-s − 0.355·17-s − 0.265·19-s − 0.667·20-s − 1.61·23-s + 0.780·25-s − 0.664·26-s − 0.587·28-s − 0.956·29-s + 0.435·31-s − 0.176·32-s + 0.251·34-s + 1.56·35-s + 0.987·37-s + 0.187·38-s + 0.471·40-s − 0.964·41-s − 0.463·43-s + 1.13·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1581418020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1581418020\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 14.9T + 125T^{2} \) |
| 7 | \( 1 + 21.7T + 343T^{2} \) |
| 13 | \( 1 - 44.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 21.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 222.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 130.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 499.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 12.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 35.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 538.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 519.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 78.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 772.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 537.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 667.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 179.T + 9.12e5T^{2} \) |
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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
−8.574674564438395993840255041830, −8.057864561307649681142780152001, −7.30818801892793323576070050002, −6.45950316281310962687582331959, −5.91128281419308798711119783029, −4.42431599465802227762226737020, −3.68132854982663211843572542101, −2.98062048436636181222562818557, −1.61372685302476062082070912499, −0.19779625247844712825986472714,
0.19779625247844712825986472714, 1.61372685302476062082070912499, 2.98062048436636181222562818557, 3.68132854982663211843572542101, 4.42431599465802227762226737020, 5.91128281419308798711119783029, 6.45950316281310962687582331959, 7.30818801892793323576070050002, 8.057864561307649681142780152001, 8.574674564438395993840255041830
