L(s) = 1 | − 2.63·2-s − 9.87·3-s − 1.07·4-s − 5·5-s + 25.9·6-s + 1.27·7-s + 23.8·8-s + 70.5·9-s + 13.1·10-s + 63.8·11-s + 10.5·12-s − 13·13-s − 3.36·14-s + 49.3·15-s − 54.2·16-s + 49.9·17-s − 185.·18-s − 23.6·19-s + 5.35·20-s − 12.6·21-s − 168.·22-s + 18.0·23-s − 235.·24-s + 25·25-s + 34.2·26-s − 430.·27-s − 1.37·28-s + ⋯ |
L(s) = 1 | − 0.930·2-s − 1.90·3-s − 0.133·4-s − 0.447·5-s + 1.76·6-s + 0.0690·7-s + 1.05·8-s + 2.61·9-s + 0.416·10-s + 1.74·11-s + 0.254·12-s − 0.277·13-s − 0.0642·14-s + 0.850·15-s − 0.848·16-s + 0.713·17-s − 2.43·18-s − 0.285·19-s + 0.0599·20-s − 0.131·21-s − 1.62·22-s + 0.163·23-s − 2.00·24-s + 0.200·25-s + 0.258·26-s − 3.06·27-s − 0.00924·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
good | 2 | \( 1 + 2.63T + 8T^{2} \) |
| 3 | \( 1 + 9.87T + 27T^{2} \) |
| 7 | \( 1 - 1.27T + 343T^{2} \) |
| 11 | \( 1 - 63.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 49.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 25.5T + 2.43e4T^{2} \) |
| 37 | \( 1 - 147.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 48.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 243.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 509.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 702.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 598.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 680.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 794.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 145.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 698.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 927.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 613.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 63.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−8.405806884897222514944038177890, −7.54335483397496697264738916165, −6.79679654907586861160274604083, −6.21332301064753249012225843853, −5.16720486920200397993875985476, −4.47452557356873187047395638259, −3.76001344898949792899346073933, −1.57026077899682105602505231398, −0.934789878222852890634874007883, 0,
0.934789878222852890634874007883, 1.57026077899682105602505231398, 3.76001344898949792899346073933, 4.47452557356873187047395638259, 5.16720486920200397993875985476, 6.21332301064753249012225843853, 6.79679654907586861160274604083, 7.54335483397496697264738916165, 8.405806884897222514944038177890