| L(s) = 1 | + 0.903·2-s + 3-s − 1.18·4-s + 3.13·5-s + 0.903·6-s + 3.44·7-s − 2.87·8-s + 9-s + 2.82·10-s − 1.18·12-s + 4.00·13-s + 3.10·14-s + 3.13·15-s − 0.227·16-s − 1.64·17-s + 0.903·18-s + 19-s − 3.70·20-s + 3.44·21-s + 0.497·23-s − 2.87·24-s + 4.80·25-s + 3.61·26-s + 27-s − 4.07·28-s − 3.13·29-s + 2.82·30-s + ⋯ |
| L(s) = 1 | + 0.638·2-s + 0.577·3-s − 0.592·4-s + 1.40·5-s + 0.368·6-s + 1.30·7-s − 1.01·8-s + 0.333·9-s + 0.893·10-s − 0.341·12-s + 1.11·13-s + 0.831·14-s + 0.808·15-s − 0.0569·16-s − 0.399·17-s + 0.212·18-s + 0.229·19-s − 0.829·20-s + 0.751·21-s + 0.103·23-s − 0.586·24-s + 0.960·25-s + 0.709·26-s + 0.192·27-s − 0.770·28-s − 0.582·29-s + 0.516·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.867382319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.867382319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.903T + 2T^{2} \) |
| 5 | \( 1 - 3.13T + 5T^{2} \) |
| 7 | \( 1 - 3.44T + 7T^{2} \) |
| 13 | \( 1 - 4.00T + 13T^{2} \) |
| 17 | \( 1 + 1.64T + 17T^{2} \) |
| 23 | \( 1 - 0.497T + 23T^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 + 5.97T + 31T^{2} \) |
| 37 | \( 1 - 9.79T + 37T^{2} \) |
| 41 | \( 1 - 4.40T + 41T^{2} \) |
| 43 | \( 1 - 4.36T + 43T^{2} \) |
| 47 | \( 1 + 3.09T + 47T^{2} \) |
| 53 | \( 1 - 9.30T + 53T^{2} \) |
| 59 | \( 1 + 6.95T + 59T^{2} \) |
| 61 | \( 1 + 8.05T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 1.86T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 3.41T + 83T^{2} \) |
| 89 | \( 1 + 6.81T + 89T^{2} \) |
| 97 | \( 1 + 4.43T + 97T^{2} \) |
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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
−8.103916368188968150369224845949, −7.31371840307810698312651258094, −6.20371276418833703005575992534, −5.79356113804937894951944374718, −5.10657151535759503789418970195, −4.39969534827284180939397035395, −3.72112710478033321076898693243, −2.71028386702752719436175843662, −1.91127705649358966812998929856, −1.09648922269214026256749235675,
1.09648922269214026256749235675, 1.91127705649358966812998929856, 2.71028386702752719436175843662, 3.72112710478033321076898693243, 4.39969534827284180939397035395, 5.10657151535759503789418970195, 5.79356113804937894951944374718, 6.20371276418833703005575992534, 7.31371840307810698312651258094, 8.103916368188968150369224845949
