L(s) = 1 | − 0.299·2-s + 3.01·3-s − 1.91·4-s + 3.67·5-s − 0.903·6-s − 0.783·7-s + 1.17·8-s + 6.10·9-s − 1.10·10-s − 1.47·11-s − 5.76·12-s − 13-s + 0.234·14-s + 11.0·15-s + 3.46·16-s − 5.71·17-s − 1.82·18-s − 1.55·19-s − 7.02·20-s − 2.36·21-s + 0.442·22-s + 4.12·23-s + 3.53·24-s + 8.51·25-s + 0.299·26-s + 9.37·27-s + 1.49·28-s + ⋯ |
L(s) = 1 | − 0.211·2-s + 1.74·3-s − 0.955·4-s + 1.64·5-s − 0.369·6-s − 0.296·7-s + 0.414·8-s + 2.03·9-s − 0.348·10-s − 0.445·11-s − 1.66·12-s − 0.277·13-s + 0.0626·14-s + 2.86·15-s + 0.867·16-s − 1.38·17-s − 0.431·18-s − 0.356·19-s − 1.57·20-s − 0.515·21-s + 0.0943·22-s + 0.860·23-s + 0.721·24-s + 1.70·25-s + 0.0587·26-s + 1.80·27-s + 0.282·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.125573570\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.125573570\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 0.299T + 2T^{2} \) |
| 3 | \( 1 - 3.01T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 + 0.783T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 17 | \( 1 + 5.71T + 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 23 | \( 1 - 4.12T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 6.37T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 7.62T + 59T^{2} \) |
| 61 | \( 1 + 5.19T + 61T^{2} \) |
| 67 | \( 1 + 2.69T + 67T^{2} \) |
| 71 | \( 1 + 5.68T + 71T^{2} \) |
| 73 | \( 1 - 9.03T + 73T^{2} \) |
| 79 | \( 1 + 3.98T + 79T^{2} \) |
| 83 | \( 1 + 0.997T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 6.88T + 97T^{2} \) |
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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
−10.79972994508412708630134820078, −9.827051263715422338893085395217, −9.351692807441014053677075641791, −8.840146597104196439587687571165, −7.87192162869237479144096174485, −6.70847664759941524155761044549, −5.34591239953407421839771722474, −4.18690033350946735162270303109, −2.82601134736842073380180582416, −1.84577727613477550226818928670,
1.84577727613477550226818928670, 2.82601134736842073380180582416, 4.18690033350946735162270303109, 5.34591239953407421839771722474, 6.70847664759941524155761044549, 7.87192162869237479144096174485, 8.840146597104196439587687571165, 9.351692807441014053677075641791, 9.827051263715422338893085395217, 10.79972994508412708630134820078