L(s) = 1 | + 3.43·3-s + 2.52·5-s + 2.67·7-s + 8.80·9-s + 1.07·11-s − 0.0741·13-s + 8.68·15-s + 2.98·17-s − 5.23·19-s + 9.18·21-s + 0.357·23-s + 1.39·25-s + 19.9·27-s − 10.3·29-s − 0.191·31-s + 3.70·33-s + 6.75·35-s + 6.37·37-s − 0.254·39-s − 1.35·41-s − 3.54·43-s + 22.2·45-s + 1.08·47-s + 0.142·49-s + 10.2·51-s − 7.34·53-s + 2.72·55-s + ⋯ |
L(s) = 1 | + 1.98·3-s + 1.13·5-s + 1.01·7-s + 2.93·9-s + 0.325·11-s − 0.0205·13-s + 2.24·15-s + 0.724·17-s − 1.20·19-s + 2.00·21-s + 0.0744·23-s + 0.278·25-s + 3.83·27-s − 1.91·29-s − 0.0344·31-s + 0.645·33-s + 1.14·35-s + 1.04·37-s − 0.0407·39-s − 0.211·41-s − 0.539·43-s + 3.31·45-s + 0.158·47-s + 0.0203·49-s + 1.43·51-s − 1.00·53-s + 0.368·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.865724632\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.865724632\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - 3.43T + 3T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 + 0.0741T + 13T^{2} \) |
| 17 | \( 1 - 2.98T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 - 0.357T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 0.191T + 31T^{2} \) |
| 37 | \( 1 - 6.37T + 37T^{2} \) |
| 41 | \( 1 + 1.35T + 41T^{2} \) |
| 43 | \( 1 + 3.54T + 43T^{2} \) |
| 47 | \( 1 - 1.08T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 - 0.128T + 61T^{2} \) |
| 67 | \( 1 - 5.46T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 5.95T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 8.71T + 83T^{2} \) |
| 89 | \( 1 + 8.31T + 89T^{2} \) |
| 97 | \( 1 - 3.53T + 97T^{2} \) |
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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
−7.83945414145118292693300546339, −7.48364102842951106115898488303, −6.58406630684228214493064273556, −5.77668435605589358942969130509, −4.82946439377833063103586922747, −4.15057158799040226436655046250, −3.41197830754262826196240917316, −2.50943166192314642849011613380, −1.85773207407179989856254259965, −1.42155599863918886311044148687,
1.42155599863918886311044148687, 1.85773207407179989856254259965, 2.50943166192314642849011613380, 3.41197830754262826196240917316, 4.15057158799040226436655046250, 4.82946439377833063103586922747, 5.77668435605589358942969130509, 6.58406630684228214493064273556, 7.48364102842951106115898488303, 7.83945414145118292693300546339