| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s − 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s − 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1213 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1213 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.581784786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.581784786\) |
| \(L(1)\) |
\(\approx\) |
\(1.077616728\) |
| \(L(1)\) |
\(\approx\) |
\(1.077616728\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1213 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.807539073363198892457364631793, −20.09375489508134491419107624958, −19.766682681423839406656285011814, −18.91608810663982405955664412766, −18.172580394987526773130301162613, −17.58604560474247185558390595647, −16.41713739968906459461610322245, −15.66833598244847564682150381954, −15.27099720061309834726272192935, −14.290609085061574657361391272510, −13.66414757811902330807725458439, −12.19587737207430926448758295539, −11.720798422865400223899491805502, −10.90191396386594566477530025308, −10.0456561866963956738917613556, −8.88715710350973639063615627617, −8.57969722724501039305631406014, −7.85811316467292255903123152931, −7.11253269827624415243642365613, −6.26889137967910535561527743730, −4.665041869283210446781535166167, −3.8297928256789533741988488828, −2.975029276538004026126068927881, −1.78471864216509188780341201765, −1.05974950457006602730956593305,
1.05974950457006602730956593305, 1.78471864216509188780341201765, 2.975029276538004026126068927881, 3.8297928256789533741988488828, 4.665041869283210446781535166167, 6.26889137967910535561527743730, 7.11253269827624415243642365613, 7.85811316467292255903123152931, 8.57969722724501039305631406014, 8.88715710350973639063615627617, 10.0456561866963956738917613556, 10.90191396386594566477530025308, 11.720798422865400223899491805502, 12.19587737207430926448758295539, 13.66414757811902330807725458439, 14.290609085061574657361391272510, 15.27099720061309834726272192935, 15.66833598244847564682150381954, 16.41713739968906459461610322245, 17.58604560474247185558390595647, 18.172580394987526773130301162613, 18.91608810663982405955664412766, 19.766682681423839406656285011814, 20.09375489508134491419107624958, 20.807539073363198892457364631793
