L(s) = 1 | + 2-s − 2.47·3-s + 4-s − 5-s − 2.47·6-s − 1.89·7-s + 8-s + 3.13·9-s − 10-s + 11-s − 2.47·12-s + 0.376·13-s − 1.89·14-s + 2.47·15-s + 16-s + 4.34·17-s + 3.13·18-s − 1.68·19-s − 20-s + 4.68·21-s + 22-s + 9.43·23-s − 2.47·24-s + 25-s + 0.376·26-s − 0.325·27-s − 1.89·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.42·3-s + 0.5·4-s − 0.447·5-s − 1.01·6-s − 0.715·7-s + 0.353·8-s + 1.04·9-s − 0.316·10-s + 0.301·11-s − 0.714·12-s + 0.104·13-s − 0.505·14-s + 0.639·15-s + 0.250·16-s + 1.05·17-s + 0.738·18-s − 0.385·19-s − 0.223·20-s + 1.02·21-s + 0.213·22-s + 1.96·23-s − 0.505·24-s + 0.200·25-s + 0.0738·26-s − 0.0625·27-s − 0.357·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.422913406\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.422913406\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 13 | \( 1 - 0.376T + 13T^{2} \) |
| 17 | \( 1 - 4.34T + 17T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 23 | \( 1 - 9.43T + 23T^{2} \) |
| 29 | \( 1 - 0.679T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 + 7.05T + 37T^{2} \) |
| 41 | \( 1 - 1.15T + 41T^{2} \) |
| 43 | \( 1 + 0.475T + 43T^{2} \) |
| 47 | \( 1 + 4.40T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 7.86T + 59T^{2} \) |
| 61 | \( 1 + 1.19T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 4.41T + 71T^{2} \) |
| 79 | \( 1 + 0.437T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.44032309167511530685788017636, −6.90733219440215748058736055284, −6.35913736444756479557124422455, −5.71968415196046852337477000901, −5.05607285039925362991216192007, −4.54598338369635212565034932771, −3.48707283870147279676223171545, −3.06760013315624334328494522327, −1.58737336420233060151579081981, −0.58901760802682566245905949948,
0.58901760802682566245905949948, 1.58737336420233060151579081981, 3.06760013315624334328494522327, 3.48707283870147279676223171545, 4.54598338369635212565034932771, 5.05607285039925362991216192007, 5.71968415196046852337477000901, 6.35913736444756479557124422455, 6.90733219440215748058736055284, 7.44032309167511530685788017636