| L(s) = 1 | + 3.77·2-s − 4.49·3-s + 6.23·4-s − 19.9·5-s − 16.9·6-s − 6.66·8-s − 6.81·9-s − 75.2·10-s − 49.7·11-s − 28.0·12-s + 13.4·13-s + 89.5·15-s − 75.0·16-s + 17·17-s − 25.7·18-s − 154.·19-s − 124.·20-s − 187.·22-s + 119.·23-s + 29.9·24-s + 272.·25-s + 50.6·26-s + 151.·27-s − 48.3·29-s + 337.·30-s + 333.·31-s − 229.·32-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 0.864·3-s + 0.779·4-s − 1.78·5-s − 1.15·6-s − 0.294·8-s − 0.252·9-s − 2.37·10-s − 1.36·11-s − 0.673·12-s + 0.286·13-s + 1.54·15-s − 1.17·16-s + 0.242·17-s − 0.336·18-s − 1.86·19-s − 1.38·20-s − 1.81·22-s + 1.08·23-s + 0.254·24-s + 2.18·25-s + 0.382·26-s + 1.08·27-s − 0.309·29-s + 2.05·30-s + 1.93·31-s − 1.26·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6834809009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6834809009\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 - 3.77T + 8T^{2} \) |
| 3 | \( 1 + 4.49T + 27T^{2} \) |
| 5 | \( 1 + 19.9T + 125T^{2} \) |
| 11 | \( 1 + 49.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.4T + 2.19e3T^{2} \) |
| 19 | \( 1 + 154.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 48.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 333.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 79.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 593.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 32.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 581.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 481.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 7.95T + 3.00e5T^{2} \) |
| 71 | \( 1 - 400.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 139.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 0.461T + 4.93e5T^{2} \) |
| 83 | \( 1 - 194.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 823.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 765.T + 9.12e5T^{2} \) |
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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
−10.31169886034458030490120410629, −8.557613843229555095303932633013, −8.146486851113525221601648169413, −6.88722314165387635718073972121, −6.20931055206811044088218141344, −4.97838829608378873728678408514, −4.70368807993229226453139297613, −3.57105389194827406018096076186, −2.75533345013699213861175443630, −0.36796787972467223662527762530,
0.36796787972467223662527762530, 2.75533345013699213861175443630, 3.57105389194827406018096076186, 4.70368807993229226453139297613, 4.97838829608378873728678408514, 6.20931055206811044088218141344, 6.88722314165387635718073972121, 8.146486851113525221601648169413, 8.557613843229555095303932633013, 10.31169886034458030490120410629
