| L(s) = 1 | − 2·2-s + 4.41·3-s + 4·4-s − 8.83·6-s − 8·8-s − 7.50·9-s − 37.0·11-s + 17.6·12-s + 47.2·13-s + 16·16-s + 37.7·17-s + 15.0·18-s − 7.60·19-s + 74.1·22-s + 148.·23-s − 35.3·24-s − 94.5·26-s − 152.·27-s − 84.5·29-s + 129.·31-s − 32·32-s − 163.·33-s − 75.5·34-s − 30.0·36-s + 148.·37-s + 15.2·38-s + 208.·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.849·3-s + 0.5·4-s − 0.600·6-s − 0.353·8-s − 0.277·9-s − 1.01·11-s + 0.424·12-s + 1.00·13-s + 0.250·16-s + 0.538·17-s + 0.196·18-s − 0.0918·19-s + 0.718·22-s + 1.34·23-s − 0.300·24-s − 0.713·26-s − 1.08·27-s − 0.541·29-s + 0.752·31-s − 0.176·32-s − 0.864·33-s − 0.380·34-s − 0.138·36-s + 0.658·37-s + 0.0649·38-s + 0.856·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.946071022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.946071022\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 4.41T + 27T^{2} \) |
| 11 | \( 1 + 37.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 47.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 7.60T + 6.85e3T^{2} \) |
| 23 | \( 1 - 148.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 84.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 148.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.16T + 6.89e4T^{2} \) |
| 43 | \( 1 - 523.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 423.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 344.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 660.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 11.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 720.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 161.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 607.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 215.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 390.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 853.T + 9.12e5T^{2} \) |
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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
−8.651915628835821556456252711100, −7.86525104107069318779336690689, −7.49936849200396899452712663658, −6.31597441846534154136437548241, −5.65031846295531341395706430647, −4.56992203871242894924821581804, −3.29254135013474120153508580891, −2.85869868673944697344844679706, −1.77517892919003664212288586663, −0.65361788453495022197497466841,
0.65361788453495022197497466841, 1.77517892919003664212288586663, 2.85869868673944697344844679706, 3.29254135013474120153508580891, 4.56992203871242894924821581804, 5.65031846295531341395706430647, 6.31597441846534154136437548241, 7.49936849200396899452712663658, 7.86525104107069318779336690689, 8.651915628835821556456252711100
