L(s) = 1 | + 0.133·2-s − 3-s − 1.98·4-s − 2.34·5-s − 0.133·6-s − 7-s − 0.532·8-s + 9-s − 0.313·10-s + 6.47·11-s + 1.98·12-s − 0.401·13-s − 0.133·14-s + 2.34·15-s + 3.89·16-s + 7.46·17-s + 0.133·18-s + 3.14·19-s + 4.65·20-s + 21-s + 0.865·22-s + 6.04·23-s + 0.532·24-s + 0.508·25-s − 0.0537·26-s − 27-s + 1.98·28-s + ⋯ |
L(s) = 1 | + 0.0945·2-s − 0.577·3-s − 0.991·4-s − 1.04·5-s − 0.0545·6-s − 0.377·7-s − 0.188·8-s + 0.333·9-s − 0.0992·10-s + 1.95·11-s + 0.572·12-s − 0.111·13-s − 0.0357·14-s + 0.606·15-s + 0.973·16-s + 1.80·17-s + 0.0315·18-s + 0.722·19-s + 1.04·20-s + 0.218·21-s + 0.184·22-s + 1.25·23-s + 0.108·24-s + 0.101·25-s − 0.0105·26-s − 0.192·27-s + 0.374·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.399303916\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.399303916\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 - 0.133T + 2T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 11 | \( 1 - 6.47T + 11T^{2} \) |
| 13 | \( 1 + 0.401T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 2.40T + 31T^{2} \) |
| 37 | \( 1 - 6.17T + 37T^{2} \) |
| 41 | \( 1 - 7.00T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 2.01T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 - 0.960T + 59T^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 + 2.40T + 67T^{2} \) |
| 71 | \( 1 + 7.25T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 1.45T + 97T^{2} \) |
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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
−7.74309108899105002743475155947, −7.21145106876475560503424828918, −6.38593492436561121846182892765, −5.73018675322097414810820716952, −4.90183645332241033285984756355, −4.29694092227580644506961780119, −3.58340137108139691566098028104, −3.15554741758135017577072740028, −1.20830631790241375311521285436, −0.74964115615536066727775095120,
0.74964115615536066727775095120, 1.20830631790241375311521285436, 3.15554741758135017577072740028, 3.58340137108139691566098028104, 4.29694092227580644506961780119, 4.90183645332241033285984756355, 5.73018675322097414810820716952, 6.38593492436561121846182892765, 7.21145106876475560503424828918, 7.74309108899105002743475155947