| L(s) = 1 | − 2-s + 2.44·3-s + 4-s − 2.44·6-s − 7-s − 8-s + 2.99·9-s − 11-s + 2.44·12-s + 4.89·13-s + 14-s + 16-s + 2·17-s − 2.99·18-s − 0.449·19-s − 2.44·21-s + 22-s − 1.55·23-s − 2.44·24-s − 4.89·26-s − 28-s + 2.44·29-s + 2.89·31-s − 32-s − 2.44·33-s − 2·34-s + 2.99·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.41·3-s + 0.5·4-s − 0.999·6-s − 0.377·7-s − 0.353·8-s + 0.999·9-s − 0.301·11-s + 0.707·12-s + 1.35·13-s + 0.267·14-s + 0.250·16-s + 0.485·17-s − 0.707·18-s − 0.103·19-s − 0.534·21-s + 0.213·22-s − 0.323·23-s − 0.499·24-s − 0.960·26-s − 0.188·28-s + 0.454·29-s + 0.520·31-s − 0.176·32-s − 0.426·33-s − 0.342·34-s + 0.499·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.362560328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.362560328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 0.449T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 - 2.89T + 31T^{2} \) |
| 37 | \( 1 - 0.449T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 0.449T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 - 9.34T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 6.89T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.524121020987614731310916233558, −7.896426463726437670132624513134, −7.42132222523749062943621344907, −6.35278727441398668597900259422, −5.80032195647310423352633670083, −4.41284644566979973199552600437, −3.54976141300519278628834220176, −2.91047953137287965425257374637, −2.06146065067009206268604693866, −0.949091867304955170959500335394,
0.949091867304955170959500335394, 2.06146065067009206268604693866, 2.91047953137287965425257374637, 3.54976141300519278628834220176, 4.41284644566979973199552600437, 5.80032195647310423352633670083, 6.35278727441398668597900259422, 7.42132222523749062943621344907, 7.896426463726437670132624513134, 8.524121020987614731310916233558
