L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 3.13·7-s + 8-s + 9-s + 10-s + 11-s − 12-s − 6.38·13-s − 3.13·14-s − 15-s + 16-s − 17-s + 18-s − 6.38·19-s + 20-s + 3.13·21-s + 22-s − 7.44·23-s − 24-s + 25-s − 6.38·26-s − 27-s − 3.13·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.18·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 1.77·13-s − 0.837·14-s − 0.258·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 1.46·19-s + 0.223·20-s + 0.683·21-s + 0.213·22-s − 1.55·23-s − 0.204·24-s + 0.200·25-s − 1.25·26-s − 0.192·27-s − 0.592·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.784441494\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.784441494\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 3.13T + 7T^{2} \) |
| 13 | \( 1 + 6.38T + 13T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 - 0.747T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 6.31T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 7.45T + 59T^{2} \) |
| 61 | \( 1 - 9.38T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 + 9.81T + 73T^{2} \) |
| 79 | \( 1 - 1.00T + 79T^{2} \) |
| 83 | \( 1 - 3.24T + 83T^{2} \) |
| 89 | \( 1 + 4.50T + 89T^{2} \) |
| 97 | \( 1 - 8.63T + 97T^{2} \) |
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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
−7.913667757363108353856218502441, −7.15717635713407081213379842203, −6.49118257203428017168671829973, −6.03160185517940722118481923263, −5.38332064606593406248346852940, −4.28260825156693366879884176754, −4.06853763284379086018596123768, −2.51823761944682185927712764537, −2.36805057963562232201636360175, −0.62179856453201123284042363970,
0.62179856453201123284042363970, 2.36805057963562232201636360175, 2.51823761944682185927712764537, 4.06853763284379086018596123768, 4.28260825156693366879884176754, 5.38332064606593406248346852940, 6.03160185517940722118481923263, 6.49118257203428017168671829973, 7.15717635713407081213379842203, 7.913667757363108353856218502441