L(s) = 1 | − 2-s + 3.40·3-s + 4-s + 2.94·5-s − 3.40·6-s + 0.502·7-s − 8-s + 8.56·9-s − 2.94·10-s − 2.01·11-s + 3.40·12-s + 4.36·13-s − 0.502·14-s + 10.0·15-s + 16-s + 3.17·17-s − 8.56·18-s + 19-s + 2.94·20-s + 1.70·21-s + 2.01·22-s + 7.38·23-s − 3.40·24-s + 3.67·25-s − 4.36·26-s + 18.9·27-s + 0.502·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.96·3-s + 0.5·4-s + 1.31·5-s − 1.38·6-s + 0.189·7-s − 0.353·8-s + 2.85·9-s − 0.931·10-s − 0.606·11-s + 0.981·12-s + 1.20·13-s − 0.134·14-s + 2.58·15-s + 0.250·16-s + 0.770·17-s − 2.01·18-s + 0.229·19-s + 0.658·20-s + 0.372·21-s + 0.428·22-s + 1.53·23-s − 0.694·24-s + 0.734·25-s − 0.855·26-s + 3.64·27-s + 0.0949·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.961523291\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.961523291\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 3.40T + 3T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 7 | \( 1 - 0.502T + 7T^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 - 4.36T + 13T^{2} \) |
| 17 | \( 1 - 3.17T + 17T^{2} \) |
| 23 | \( 1 - 7.38T + 23T^{2} \) |
| 29 | \( 1 + 3.10T + 29T^{2} \) |
| 31 | \( 1 + 9.26T + 31T^{2} \) |
| 37 | \( 1 + 0.509T + 37T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 - 7.27T + 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 1.55T + 61T^{2} \) |
| 67 | \( 1 + 2.06T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 6.75T + 73T^{2} \) |
| 79 | \( 1 + 3.27T + 79T^{2} \) |
| 83 | \( 1 + 8.58T + 83T^{2} \) |
| 89 | \( 1 + 7.50T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−8.062420465396861917266206101564, −7.27384049105197848572223997396, −6.85939864375724368609072872223, −5.78008058795652843074080127277, −5.14273570359517398078888896030, −3.92029231886170090840766768955, −3.19515923185358651136586684984, −2.61905543169216383192369619278, −1.68368804722455377925862020184, −1.33175223643325115253735092747,
1.33175223643325115253735092747, 1.68368804722455377925862020184, 2.61905543169216383192369619278, 3.19515923185358651136586684984, 3.92029231886170090840766768955, 5.14273570359517398078888896030, 5.78008058795652843074080127277, 6.85939864375724368609072872223, 7.27384049105197848572223997396, 8.062420465396861917266206101564