| L(s) = 1 | − 1.64·2-s − 1.94·3-s + 0.704·4-s − 2.95·5-s + 3.19·6-s + 4.77·7-s + 2.13·8-s + 0.774·9-s + 4.86·10-s + 11-s − 1.36·12-s − 2.28·13-s − 7.85·14-s + 5.74·15-s − 4.91·16-s + 17-s − 1.27·18-s − 5.82·19-s − 2.08·20-s − 9.27·21-s − 1.64·22-s − 3.60·23-s − 4.13·24-s + 3.74·25-s + 3.76·26-s + 4.32·27-s + 3.36·28-s + ⋯ |
| L(s) = 1 | − 1.16·2-s − 1.12·3-s + 0.352·4-s − 1.32·5-s + 1.30·6-s + 1.80·7-s + 0.753·8-s + 0.258·9-s + 1.53·10-s + 0.301·11-s − 0.394·12-s − 0.635·13-s − 2.09·14-s + 1.48·15-s − 1.22·16-s + 0.242·17-s − 0.300·18-s − 1.33·19-s − 0.465·20-s − 2.02·21-s − 0.350·22-s − 0.751·23-s − 0.845·24-s + 0.748·25-s + 0.738·26-s + 0.832·27-s + 0.635·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1727120839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1727120839\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.64T + 2T^{2} \) |
| 3 | \( 1 + 1.94T + 3T^{2} \) |
| 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 13 | \( 1 + 2.28T + 13T^{2} \) |
| 19 | \( 1 + 5.82T + 19T^{2} \) |
| 23 | \( 1 + 3.60T + 23T^{2} \) |
| 29 | \( 1 + 5.58T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 + 9.62T + 41T^{2} \) |
| 47 | \( 1 - 6.14T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 - 9.16T + 59T^{2} \) |
| 61 | \( 1 + 2.77T + 61T^{2} \) |
| 67 | \( 1 - 6.69T + 67T^{2} \) |
| 71 | \( 1 + 5.95T + 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 79 | \( 1 + 4.03T + 79T^{2} \) |
| 83 | \( 1 + 3.03T + 83T^{2} \) |
| 89 | \( 1 + 3.24T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.923675617469024448760948401577, −7.28127644916401227085090607192, −6.86059350856461759408576632076, −5.53057579766747537088322807865, −5.15516424556239336015852285745, −4.25809274595147803954640309422, −3.91656893687680824630897556143, −2.16849985496022815554575458309, −1.43237316676323678188663485922, −0.27780894785439689580004075811,
0.27780894785439689580004075811, 1.43237316676323678188663485922, 2.16849985496022815554575458309, 3.91656893687680824630897556143, 4.25809274595147803954640309422, 5.15516424556239336015852285745, 5.53057579766747537088322807865, 6.86059350856461759408576632076, 7.28127644916401227085090607192, 7.923675617469024448760948401577
