| L(s) = 1 | − 1.14·2-s − 3.37·3-s − 0.698·4-s + 2.91·5-s + 3.85·6-s + 1.13·7-s + 3.07·8-s + 8.41·9-s − 3.32·10-s − 3.36·11-s + 2.36·12-s − 1.29·14-s − 9.85·15-s − 2.11·16-s − 2.07·17-s − 9.59·18-s + 6.06·19-s − 2.03·20-s − 3.84·21-s + 3.83·22-s − 5.19·23-s − 10.3·24-s + 3.51·25-s − 18.2·27-s − 0.796·28-s + 6.22·29-s + 11.2·30-s + ⋯ |
| L(s) = 1 | − 0.806·2-s − 1.95·3-s − 0.349·4-s + 1.30·5-s + 1.57·6-s + 0.430·7-s + 1.08·8-s + 2.80·9-s − 1.05·10-s − 1.01·11-s + 0.681·12-s − 0.347·14-s − 2.54·15-s − 0.528·16-s − 0.503·17-s − 2.26·18-s + 1.39·19-s − 0.455·20-s − 0.840·21-s + 0.818·22-s − 1.08·23-s − 2.12·24-s + 0.702·25-s − 3.51·27-s − 0.150·28-s + 1.15·29-s + 2.05·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6954234035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6954234035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.14T + 2T^{2} \) |
| 3 | \( 1 + 3.37T + 3T^{2} \) |
| 5 | \( 1 - 2.91T + 5T^{2} \) |
| 7 | \( 1 - 1.13T + 7T^{2} \) |
| 11 | \( 1 + 3.36T + 11T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 - 6.06T + 19T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 - 6.22T + 29T^{2} \) |
| 37 | \( 1 - 9.20T + 37T^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 + 8.90T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 - 0.413T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 - 0.648T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 2.56T + 71T^{2} \) |
| 73 | \( 1 - 3.00T + 73T^{2} \) |
| 79 | \( 1 + 0.926T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 8.11T + 97T^{2} \) |
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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
−7.997516351057315497988717008139, −7.58464489661909999563294214698, −6.59147023231109249819141834036, −5.99953960413063680735961808209, −5.30216610375876445356519969218, −4.90754595819329722320729918363, −4.13295876756624954460810976441, −2.37270331640660874765849462247, −1.40043400741320958787880009008, −0.63285505491546139465937879118,
0.63285505491546139465937879118, 1.40043400741320958787880009008, 2.37270331640660874765849462247, 4.13295876756624954460810976441, 4.90754595819329722320729918363, 5.30216610375876445356519969218, 5.99953960413063680735961808209, 6.59147023231109249819141834036, 7.58464489661909999563294214698, 7.997516351057315497988717008139
