L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 11-s − 12-s + 4·13-s − 2·14-s + 16-s − 17-s − 18-s − 6·19-s − 2·21-s + 22-s + 2·23-s + 24-s − 4·26-s − 27-s + 2·28-s − 6·29-s − 4·31-s − 32-s + 33-s + 34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.37·19-s − 0.436·21-s + 0.213·22-s + 0.417·23-s + 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.53568978244061, −14.96782876324160, −14.75721092542103, −13.76967696274434, −13.33403091316462, −12.75004859096814, −12.21153283282117, −11.49829738477923, −10.97451691145459, −10.87348044448557, −10.20619328986919, −9.482036780898663, −8.872411903636871, −8.381934357368527, −7.928120113089759, −7.161019265198485, −6.670690261512919, −6.026490550176176, −5.451529226329498, −4.821158757374697, −4.017845472888232, −3.457900484380614, −2.298470404946540, −1.798620537922429, −0.9618785236066749, 0,
0.9618785236066749, 1.798620537922429, 2.298470404946540, 3.457900484380614, 4.017845472888232, 4.821158757374697, 5.451529226329498, 6.026490550176176, 6.670690261512919, 7.161019265198485, 7.928120113089759, 8.381934357368527, 8.872411903636871, 9.482036780898663, 10.20619328986919, 10.87348044448557, 10.97451691145459, 11.49829738477923, 12.21153283282117, 12.75004859096814, 13.33403091316462, 13.76967696274434, 14.75721092542103, 14.96782876324160, 15.53568978244061