| L(s) = 1 | + 2-s − 3-s + 4-s + 3.97·5-s − 6-s − 1.87·7-s + 8-s + 9-s + 3.97·10-s + 11-s − 12-s + 0.755·13-s − 1.87·14-s − 3.97·15-s + 16-s − 2.84·17-s + 18-s − 0.849·19-s + 3.97·20-s + 1.87·21-s + 22-s + 7.44·23-s − 24-s + 10.7·25-s + 0.755·26-s − 27-s − 1.87·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.77·5-s − 0.408·6-s − 0.708·7-s + 0.353·8-s + 0.333·9-s + 1.25·10-s + 0.301·11-s − 0.288·12-s + 0.209·13-s − 0.501·14-s − 1.02·15-s + 0.250·16-s − 0.690·17-s + 0.235·18-s − 0.194·19-s + 0.888·20-s + 0.409·21-s + 0.213·22-s + 1.55·23-s − 0.204·24-s + 2.15·25-s + 0.148·26-s − 0.192·27-s − 0.354·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.498306102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.498306102\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 3.97T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 13 | \( 1 - 0.755T + 13T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 19 | \( 1 + 0.849T + 19T^{2} \) |
| 23 | \( 1 - 7.44T + 23T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 + 3.73T + 31T^{2} \) |
| 37 | \( 1 - 9.94T + 37T^{2} \) |
| 41 | \( 1 - 6.29T + 41T^{2} \) |
| 43 | \( 1 + 2.11T + 43T^{2} \) |
| 47 | \( 1 + 0.984T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 8.16T + 59T^{2} \) |
| 67 | \( 1 + 3.62T + 67T^{2} \) |
| 71 | \( 1 + 8.61T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 8.83T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 7.02T + 89T^{2} \) |
| 97 | \( 1 - 5.62T + 97T^{2} \) |
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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
−8.638938658435757048820998254338, −7.27273785181690623299008393564, −6.67727306851639967075232184815, −6.12921401706379596192233025370, −5.57111917011034579127011079970, −4.89820870246500834362846844918, −3.94282062098976836028335362510, −2.85255054181356782113057947793, −2.10955631557434139874069906457, −1.04001117995748847437104918688,
1.04001117995748847437104918688, 2.10955631557434139874069906457, 2.85255054181356782113057947793, 3.94282062098976836028335362510, 4.89820870246500834362846844918, 5.57111917011034579127011079970, 6.12921401706379596192233025370, 6.67727306851639967075232184815, 7.27273785181690623299008393564, 8.638938658435757048820998254338
