L(s) = 1 | − 2-s − 0.600·3-s + 4-s − 5-s + 0.600·6-s − 1.43·7-s − 8-s − 2.63·9-s + 10-s − 2.77·11-s − 0.600·12-s + 1.43·14-s + 0.600·15-s + 16-s − 4.50·17-s + 2.63·18-s − 4.33·19-s − 20-s + 0.863·21-s + 2.77·22-s + 4.43·23-s + 0.600·24-s + 25-s + 3.38·27-s − 1.43·28-s + 7.86·29-s − 0.600·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.346·3-s + 0.5·4-s − 0.447·5-s + 0.245·6-s − 0.544·7-s − 0.353·8-s − 0.879·9-s + 0.316·10-s − 0.835·11-s − 0.173·12-s + 0.384·14-s + 0.154·15-s + 0.250·16-s − 1.09·17-s + 0.622·18-s − 0.993·19-s − 0.223·20-s + 0.188·21-s + 0.590·22-s + 0.925·23-s + 0.122·24-s + 0.200·25-s + 0.651·27-s − 0.272·28-s + 1.46·29-s − 0.109·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4904420565\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4904420565\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 0.600T + 3T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 2.77T + 11T^{2} \) |
| 17 | \( 1 + 4.50T + 17T^{2} \) |
| 19 | \( 1 + 4.33T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 4.16T + 31T^{2} \) |
| 37 | \( 1 - 0.575T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 - 2.61T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 9.57T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 5.32T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 1.66T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 5.95T + 89T^{2} \) |
| 97 | \( 1 - 1.58T + 97T^{2} \) |
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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
−9.073882657010949610839928074859, −8.711224442218591205138761534850, −7.87630928753447718924660743285, −6.94260808330289039954190153412, −6.31797009907796571059022816974, −5.38423389897316425317755184682, −4.39394589060347104651760362782, −3.11613844179380137268037887508, −2.32725121091179356056522994293, −0.51421996707459224480775320660,
0.51421996707459224480775320660, 2.32725121091179356056522994293, 3.11613844179380137268037887508, 4.39394589060347104651760362782, 5.38423389897316425317755184682, 6.31797009907796571059022816974, 6.94260808330289039954190153412, 7.87630928753447718924660743285, 8.711224442218591205138761534850, 9.073882657010949610839928074859