L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 4.89·7-s − 8-s + 9-s − 10-s − 12-s + 6.89·13-s + 4.89·14-s − 15-s + 16-s − 17-s − 18-s + 4·19-s + 20-s + 4.89·21-s − 4·23-s + 24-s + 25-s − 6.89·26-s − 27-s − 4.89·28-s + 6·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.85·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.288·12-s + 1.91·13-s + 1.30·14-s − 0.258·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 1.06·21-s − 0.834·23-s + 0.204·24-s + 0.200·25-s − 1.35·26-s − 0.192·27-s − 0.925·28-s + 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7949796362\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7949796362\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 - 8.89T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 0.898T + 67T^{2} \) |
| 71 | \( 1 + 8.89T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 5.79T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 7.79T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−10.66897452450089206237561704530, −9.994431475052425781178676731231, −9.306235401155395774951580444631, −8.406882925152879161715122905040, −7.08122970200987101916578871551, −6.21828911469507733362298839674, −5.84868541034693098454946060810, −3.96246986274276311927666544286, −2.82239410900099257660819290117, −0.943286290071500998448808580284,
0.943286290071500998448808580284, 2.82239410900099257660819290117, 3.96246986274276311927666544286, 5.84868541034693098454946060810, 6.21828911469507733362298839674, 7.08122970200987101916578871551, 8.406882925152879161715122905040, 9.306235401155395774951580444631, 9.994431475052425781178676731231, 10.66897452450089206237561704530