L(s) = 1 | + 1.25·2-s + 3.01·3-s − 0.420·4-s + 5-s + 3.78·6-s + 3.72·7-s − 3.04·8-s + 6.07·9-s + 1.25·10-s − 3.35·11-s − 1.26·12-s + 4.84·13-s + 4.68·14-s + 3.01·15-s − 2.98·16-s − 2.67·17-s + 7.63·18-s − 0.420·20-s + 11.2·21-s − 4.21·22-s + 1.87·23-s − 9.16·24-s + 25-s + 6.08·26-s + 9.25·27-s − 1.56·28-s − 5.25·29-s + ⋯ |
L(s) = 1 | + 0.888·2-s + 1.73·3-s − 0.210·4-s + 0.447·5-s + 1.54·6-s + 1.40·7-s − 1.07·8-s + 2.02·9-s + 0.397·10-s − 1.01·11-s − 0.365·12-s + 1.34·13-s + 1.25·14-s + 0.777·15-s − 0.745·16-s − 0.648·17-s + 1.79·18-s − 0.0939·20-s + 2.44·21-s − 0.899·22-s + 0.391·23-s − 1.87·24-s + 0.200·25-s + 1.19·26-s + 1.78·27-s − 0.295·28-s − 0.975·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.214323994\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.214323994\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 1.25T + 2T^{2} \) |
| 3 | \( 1 - 3.01T + 3T^{2} \) |
| 7 | \( 1 - 3.72T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 + 5.25T + 29T^{2} \) |
| 31 | \( 1 + 3.11T + 31T^{2} \) |
| 37 | \( 1 - 0.992T + 37T^{2} \) |
| 41 | \( 1 - 0.416T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 - 0.260T + 53T^{2} \) |
| 59 | \( 1 + 5.18T + 59T^{2} \) |
| 61 | \( 1 - 0.768T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 2.00T + 71T^{2} \) |
| 73 | \( 1 - 4.56T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 7.66T + 89T^{2} \) |
| 97 | \( 1 + 7.22T + 97T^{2} \) |
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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
−8.994056854479075727318110829418, −8.495806698488258729633478262984, −7.986037956034398475449387335607, −7.02099199282957293673146655851, −5.78948110954744834623506274958, −4.98916408629681730092389742713, −4.19162828505412822657127847469, −3.41385222898897442412676439785, −2.48219577560886949315594534746, −1.59039481249362113921164222573,
1.59039481249362113921164222573, 2.48219577560886949315594534746, 3.41385222898897442412676439785, 4.19162828505412822657127847469, 4.98916408629681730092389742713, 5.78948110954744834623506274958, 7.02099199282957293673146655851, 7.986037956034398475449387335607, 8.495806698488258729633478262984, 8.994056854479075727318110829418