L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 0.459·11-s − 12-s + 5.04·13-s − 14-s + 15-s + 16-s − 7.89·17-s − 18-s + 7.50·19-s − 20-s − 21-s − 0.459·22-s − 23-s + 24-s + 25-s − 5.04·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.138·11-s − 0.288·12-s + 1.39·13-s − 0.267·14-s + 0.258·15-s + 0.250·16-s − 1.91·17-s − 0.235·18-s + 1.72·19-s − 0.223·20-s − 0.218·21-s − 0.0980·22-s − 0.208·23-s + 0.204·24-s + 0.200·25-s − 0.989·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.080406009\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.080406009\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 0.459T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 17 | \( 1 + 7.89T + 17T^{2} \) |
| 19 | \( 1 - 7.50T + 19T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 31 | \( 1 + 0.700T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 7.74T + 43T^{2} \) |
| 47 | \( 1 - 2.58T + 47T^{2} \) |
| 53 | \( 1 - 3.19T + 53T^{2} \) |
| 59 | \( 1 + 8.20T + 59T^{2} \) |
| 61 | \( 1 - 1.08T + 61T^{2} \) |
| 67 | \( 1 - 8.66T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 6.11T + 79T^{2} \) |
| 83 | \( 1 - 4.42T + 83T^{2} \) |
| 89 | \( 1 + 3.16T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−8.222635149591157042889463653142, −7.63757334213686099698709526241, −6.89226180685488209496437119788, −6.20163840531391873879962545740, −5.52900135874176643879929231323, −4.50224288159631936531310213455, −3.84270342275831887603976702009, −2.75112365791650512854379488529, −1.59871872875230416855846532439, −0.69121207935197451547832028142,
0.69121207935197451547832028142, 1.59871872875230416855846532439, 2.75112365791650512854379488529, 3.84270342275831887603976702009, 4.50224288159631936531310213455, 5.52900135874176643879929231323, 6.20163840531391873879962545740, 6.89226180685488209496437119788, 7.63757334213686099698709526241, 8.222635149591157042889463653142