| L(s) = 1 | − 2-s − 1.00·3-s + 4-s − 2.36·5-s + 1.00·6-s + 3.55·7-s − 8-s − 1.99·9-s + 2.36·10-s − 6.50·11-s − 1.00·12-s − 3.51·13-s − 3.55·14-s + 2.36·15-s + 16-s − 4.49·17-s + 1.99·18-s + 0.0970·19-s − 2.36·20-s − 3.55·21-s + 6.50·22-s + 0.893·23-s + 1.00·24-s + 0.584·25-s + 3.51·26-s + 5.00·27-s + 3.55·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.05·5-s + 0.408·6-s + 1.34·7-s − 0.353·8-s − 0.666·9-s + 0.747·10-s − 1.96·11-s − 0.288·12-s − 0.973·13-s − 0.950·14-s + 0.610·15-s + 0.250·16-s − 1.08·17-s + 0.471·18-s + 0.0222·19-s − 0.528·20-s − 0.776·21-s + 1.38·22-s + 0.186·23-s + 0.204·24-s + 0.116·25-s + 0.688·26-s + 0.962·27-s + 0.672·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3060879347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3060879347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 1.00T + 3T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 7 | \( 1 - 3.55T + 7T^{2} \) |
| 11 | \( 1 + 6.50T + 11T^{2} \) |
| 13 | \( 1 + 3.51T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 - 0.0970T + 19T^{2} \) |
| 23 | \( 1 - 0.893T + 23T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 + 2.91T + 31T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 9.93T + 43T^{2} \) |
| 47 | \( 1 + 5.67T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 - 6.52T + 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 - 1.05T + 71T^{2} \) |
| 73 | \( 1 + 1.36T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 1.41T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−8.511326050610524226270224461136, −8.064879057559153574383036277929, −7.57571532310100499912667371329, −6.82047425081843011313504369519, −5.49548828881072132330875329839, −5.11599979865726979107748041139, −4.23572037677867163652754126442, −2.84679545903238040134547199778, −2.06933402796457244771699119339, −0.36865679993030449238010079964,
0.36865679993030449238010079964, 2.06933402796457244771699119339, 2.84679545903238040134547199778, 4.23572037677867163652754126442, 5.11599979865726979107748041139, 5.49548828881072132330875329839, 6.82047425081843011313504369519, 7.57571532310100499912667371329, 8.064879057559153574383036277929, 8.511326050610524226270224461136
