| L(s) = 1 | + 2-s + 2.67·3-s + 4-s − 2.78·5-s + 2.67·6-s − 7-s + 8-s + 4.15·9-s − 2.78·10-s − 4.75·11-s + 2.67·12-s + 0.615·13-s − 14-s − 7.45·15-s + 16-s + 7.22·17-s + 4.15·18-s − 2.78·20-s − 2.67·21-s − 4.75·22-s + 3.07·23-s + 2.67·24-s + 2.77·25-s + 0.615·26-s + 3.10·27-s − 28-s + 0.676·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.54·3-s + 0.5·4-s − 1.24·5-s + 1.09·6-s − 0.377·7-s + 0.353·8-s + 1.38·9-s − 0.881·10-s − 1.43·11-s + 0.772·12-s + 0.170·13-s − 0.267·14-s − 1.92·15-s + 0.250·16-s + 1.75·17-s + 0.980·18-s − 0.623·20-s − 0.583·21-s − 1.01·22-s + 0.641·23-s + 0.546·24-s + 0.554·25-s + 0.120·26-s + 0.596·27-s − 0.188·28-s + 0.125·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.194342674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.194342674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.67T + 3T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 - 0.615T + 13T^{2} \) |
| 17 | \( 1 - 7.22T + 17T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 - 0.676T + 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 - 7.74T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 + 4.03T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 7.48T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 + 0.372T + 71T^{2} \) |
| 73 | \( 1 + 8.38T + 73T^{2} \) |
| 79 | \( 1 - 6.66T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.174549952939939845795584543015, −7.45837942058311417969993459439, −7.29440386476798224674080541726, −5.94884055744290410165885722315, −5.19029454554883315881374768142, −4.16666983585830311100618183994, −3.70709811041578107800837149336, −2.85640936389558255328026863056, −2.55509066950608697425105683597, −0.939904348303023576184099178728,
0.939904348303023576184099178728, 2.55509066950608697425105683597, 2.85640936389558255328026863056, 3.70709811041578107800837149336, 4.16666983585830311100618183994, 5.19029454554883315881374768142, 5.94884055744290410165885722315, 7.29440386476798224674080541726, 7.45837942058311417969993459439, 8.174549952939939845795584543015
