| L(s) = 1 | + 2.52·2-s − 2.46·3-s + 4.37·4-s + 5-s − 6.22·6-s + 2.09·7-s + 5.99·8-s + 3.08·9-s + 2.52·10-s − 10.7·12-s + 1.28·13-s + 5.28·14-s − 2.46·15-s + 6.37·16-s − 2.99·17-s + 7.78·18-s + 2.15·19-s + 4.37·20-s − 5.16·21-s + 8.77·23-s − 14.7·24-s + 25-s + 3.25·26-s − 0.209·27-s + 9.15·28-s − 0.612·29-s − 6.22·30-s + ⋯ |
| L(s) = 1 | + 1.78·2-s − 1.42·3-s + 2.18·4-s + 0.447·5-s − 2.54·6-s + 0.791·7-s + 2.11·8-s + 1.02·9-s + 0.798·10-s − 3.11·12-s + 0.357·13-s + 1.41·14-s − 0.636·15-s + 1.59·16-s − 0.725·17-s + 1.83·18-s + 0.493·19-s + 0.977·20-s − 1.12·21-s + 1.83·23-s − 3.01·24-s + 0.200·25-s + 0.637·26-s − 0.0402·27-s + 1.73·28-s − 0.113·29-s − 1.13·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.109161099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.109161099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 3 | \( 1 + 2.46T + 3T^{2} \) |
| 7 | \( 1 - 2.09T + 7T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 + 2.99T + 17T^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 23 | \( 1 - 8.77T + 23T^{2} \) |
| 29 | \( 1 + 0.612T + 29T^{2} \) |
| 31 | \( 1 + 3.64T + 31T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 41 | \( 1 + 5.10T + 41T^{2} \) |
| 43 | \( 1 - 5.17T + 43T^{2} \) |
| 47 | \( 1 + 7.30T + 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 + 6.49T + 59T^{2} \) |
| 61 | \( 1 + 0.502T + 61T^{2} \) |
| 67 | \( 1 + 7.80T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 5.35T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 4.32T + 89T^{2} \) |
| 97 | \( 1 + 0.351T + 97T^{2} \) |
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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
−11.18143208265521305020907431642, −10.43637538576336992251536046677, −8.949786338529673930282991699749, −7.37948948187619310636626378580, −6.58529880887731459114525705933, −5.82211887530809714242873864901, −5.05390119079528486129013001736, −4.56782242226678921392832011731, −3.13269866437963588658453419520, −1.58891951458805395883110651479,
1.58891951458805395883110651479, 3.13269866437963588658453419520, 4.56782242226678921392832011731, 5.05390119079528486129013001736, 5.82211887530809714242873864901, 6.58529880887731459114525705933, 7.37948948187619310636626378580, 8.949786338529673930282991699749, 10.43637538576336992251536046677, 11.18143208265521305020907431642
