| L(s) = 1 | − 5.18·2-s − 5.95·3-s + 18.8·4-s + 18.5·5-s + 30.8·6-s − 56.4·8-s + 8.44·9-s − 96.4·10-s − 4.03·11-s − 112.·12-s − 3.74·13-s − 110.·15-s + 141.·16-s − 31.7·17-s − 43.8·18-s − 128.·19-s + 351.·20-s + 20.9·22-s − 83.0·23-s + 336.·24-s + 220.·25-s + 19.4·26-s + 110.·27-s + 101.·29-s + 574.·30-s − 19.6·31-s − 282.·32-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 1.14·3-s + 2.36·4-s + 1.66·5-s + 2.10·6-s − 2.49·8-s + 0.312·9-s − 3.04·10-s − 0.110·11-s − 2.70·12-s − 0.0798·13-s − 1.90·15-s + 2.21·16-s − 0.453·17-s − 0.573·18-s − 1.55·19-s + 3.92·20-s + 0.202·22-s − 0.753·23-s + 2.85·24-s + 1.76·25-s + 0.146·26-s + 0.787·27-s + 0.648·29-s + 3.49·30-s − 0.113·31-s − 1.56·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - 41T \) |
| good | 2 | \( 1 + 5.18T + 8T^{2} \) |
| 3 | \( 1 + 5.95T + 27T^{2} \) |
| 5 | \( 1 - 18.5T + 125T^{2} \) |
| 11 | \( 1 + 4.03T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.74T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 83.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 101.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 19.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 408.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 86.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 389.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 568.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 253.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 483.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 751.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 287.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 508.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 265.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 847.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 962.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 408.T + 9.12e5T^{2} \) |
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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
−8.630608714416956171646508842729, −7.78165158715964239039742770773, −6.60216773517909198475374233942, −6.31671382074555293811609498860, −5.72211645683558205217607054755, −4.60129365502316154595038878149, −2.62983467335810705923714868653, −1.99917382809476205998060904757, −1.00025787394891265473152879447, 0,
1.00025787394891265473152879447, 1.99917382809476205998060904757, 2.62983467335810705923714868653, 4.60129365502316154595038878149, 5.72211645683558205217607054755, 6.31671382074555293811609498860, 6.60216773517909198475374233942, 7.78165158715964239039742770773, 8.630608714416956171646508842729
