L(s) = 1 | − 2-s + 1.92·3-s + 4-s − 0.949·5-s − 1.92·6-s − 5.12·7-s − 8-s + 0.724·9-s + 0.949·10-s + 5.68·11-s + 1.92·12-s + 1.19·13-s + 5.12·14-s − 1.83·15-s + 16-s + 1.38·17-s − 0.724·18-s − 19-s − 0.949·20-s − 9.88·21-s − 5.68·22-s − 0.766·23-s − 1.92·24-s − 4.09·25-s − 1.19·26-s − 4.39·27-s − 5.12·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.11·3-s + 0.5·4-s − 0.424·5-s − 0.787·6-s − 1.93·7-s − 0.353·8-s + 0.241·9-s + 0.300·10-s + 1.71·11-s + 0.557·12-s + 0.330·13-s + 1.36·14-s − 0.473·15-s + 0.250·16-s + 0.335·17-s − 0.170·18-s − 0.229·19-s − 0.212·20-s − 2.15·21-s − 1.21·22-s − 0.159·23-s − 0.393·24-s − 0.819·25-s − 0.233·26-s − 0.845·27-s − 0.967·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 1.92T + 3T^{2} \) |
| 5 | \( 1 + 0.949T + 5T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 - 5.68T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 23 | \( 1 + 0.766T + 23T^{2} \) |
| 29 | \( 1 - 1.61T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 + 2.57T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 8.05T + 47T^{2} \) |
| 53 | \( 1 + 3.78T + 53T^{2} \) |
| 59 | \( 1 - 3.49T + 59T^{2} \) |
| 61 | \( 1 + 6.12T + 61T^{2} \) |
| 67 | \( 1 - 9.03T + 67T^{2} \) |
| 71 | \( 1 + 1.73T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 4.84T + 79T^{2} \) |
| 83 | \( 1 + 0.862T + 83T^{2} \) |
| 89 | \( 1 - 4.83T + 89T^{2} \) |
| 97 | \( 1 + 1.09T + 97T^{2} \) |
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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
−7.59264000450247136296173139211, −6.89042713503851427857872153313, −6.30399709332378243094759608280, −5.80688854561255363284717063483, −4.16103738878601922862095021801, −3.68310804799235638580018438321, −3.13305153569889748827134896498, −2.34034139152775178829200697550, −1.20964717286042176384681077076, 0,
1.20964717286042176384681077076, 2.34034139152775178829200697550, 3.13305153569889748827134896498, 3.68310804799235638580018438321, 4.16103738878601922862095021801, 5.80688854561255363284717063483, 6.30399709332378243094759608280, 6.89042713503851427857872153313, 7.59264000450247136296173139211