| L(s) = 1 | − 1.77·2-s + 1.16·4-s − 0.742·5-s + 3.55·7-s + 1.48·8-s + 1.32·10-s + 2.30·11-s − 4.43·13-s − 6.32·14-s − 4.97·16-s + 1.82·17-s − 4.78·19-s − 0.865·20-s − 4.10·22-s + 3.20·23-s − 4.44·25-s + 7.89·26-s + 4.14·28-s + 10.2·29-s − 6.12·31-s + 5.88·32-s − 3.24·34-s − 2.63·35-s + 8.55·37-s + 8.51·38-s − 1.10·40-s − 2.95·41-s + ⋯ |
| L(s) = 1 | − 1.25·2-s + 0.583·4-s − 0.331·5-s + 1.34·7-s + 0.524·8-s + 0.417·10-s + 0.695·11-s − 1.23·13-s − 1.69·14-s − 1.24·16-s + 0.441·17-s − 1.09·19-s − 0.193·20-s − 0.874·22-s + 0.668·23-s − 0.889·25-s + 1.54·26-s + 0.783·28-s + 1.90·29-s − 1.10·31-s + 1.04·32-s − 0.556·34-s − 0.445·35-s + 1.40·37-s + 1.38·38-s − 0.173·40-s − 0.461·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9570250987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9570250987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 5 | \( 1 + 0.742T + 5T^{2} \) |
| 7 | \( 1 - 3.55T + 7T^{2} \) |
| 11 | \( 1 - 2.30T + 11T^{2} \) |
| 13 | \( 1 + 4.43T + 13T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 19 | \( 1 + 4.78T + 19T^{2} \) |
| 23 | \( 1 - 3.20T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 - 8.55T + 37T^{2} \) |
| 41 | \( 1 + 2.95T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 - 0.237T + 47T^{2} \) |
| 53 | \( 1 - 0.970T + 53T^{2} \) |
| 59 | \( 1 - 3.07T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 4.04T + 67T^{2} \) |
| 71 | \( 1 - 9.80T + 71T^{2} \) |
| 73 | \( 1 + 9.11T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 0.992T + 89T^{2} \) |
| 97 | \( 1 - 1.71T + 97T^{2} \) |
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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
−8.310160068963938849227191477571, −8.008127138232925548640711241324, −7.24272330658531233187265493002, −6.61622792463381751230640437029, −5.37387549111511384895372025762, −4.61851643916723984337358344273, −4.04158630709888764152749459776, −2.53265054305483493241361015342, −1.70545428049899688755964570267, −0.70622380810238026185860549494,
0.70622380810238026185860549494, 1.70545428049899688755964570267, 2.53265054305483493241361015342, 4.04158630709888764152749459776, 4.61851643916723984337358344273, 5.37387549111511384895372025762, 6.61622792463381751230640437029, 7.24272330658531233187265493002, 8.008127138232925548640711241324, 8.310160068963938849227191477571
