| L(s) = 1 | − 5.48·2-s + 22.1·4-s + 14.0·5-s + 24.2·7-s − 77.5·8-s − 77.3·10-s + 3.10·11-s − 13·13-s − 133.·14-s + 248.·16-s + 43.9·17-s − 85.8·19-s + 311.·20-s − 17.0·22-s + 203.·23-s + 73.4·25-s + 71.3·26-s + 537.·28-s − 31.0·29-s − 135.·31-s − 744.·32-s − 241.·34-s + 341.·35-s + 290.·37-s + 471.·38-s − 1.09e3·40-s − 148.·41-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 2.76·4-s + 1.25·5-s + 1.31·7-s − 3.42·8-s − 2.44·10-s + 0.0851·11-s − 0.277·13-s − 2.54·14-s + 3.88·16-s + 0.626·17-s − 1.03·19-s + 3.48·20-s − 0.165·22-s + 1.84·23-s + 0.587·25-s + 0.538·26-s + 3.62·28-s − 0.198·29-s − 0.786·31-s − 4.11·32-s − 1.21·34-s + 1.65·35-s + 1.28·37-s + 2.01·38-s − 4.31·40-s − 0.566·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9749658769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9749658769\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 5.48T + 8T^{2} \) |
| 5 | \( 1 - 14.0T + 125T^{2} \) |
| 7 | \( 1 - 24.2T + 343T^{2} \) |
| 11 | \( 1 - 3.10T + 1.33e3T^{2} \) |
| 17 | \( 1 - 43.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 85.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 203.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 31.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 148.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 225.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 41.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 499.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 503.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 946.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 674.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 59.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 879.T + 9.12e5T^{2} \) |
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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
−12.80342774325015878929135809529, −11.40004291641673227350565587982, −10.72974287449396841923109538428, −9.723349098011715884915672436369, −8.888707946839727007921249446855, −7.902742700429722967954113724811, −6.76251208983625190013642119656, −5.46475534796872290943980054497, −2.37676234701599773990078476292, −1.27146965433305099623107560549,
1.27146965433305099623107560549, 2.37676234701599773990078476292, 5.46475534796872290943980054497, 6.76251208983625190013642119656, 7.902742700429722967954113724811, 8.888707946839727007921249446855, 9.723349098011715884915672436369, 10.72974287449396841923109538428, 11.40004291641673227350565587982, 12.80342774325015878929135809529
