| L(s) = 1 | + 2-s + 4-s − 1.44·5-s − 7-s + 8-s − 1.44·10-s + 2·11-s + 4.89·13-s − 14-s + 16-s + 2·17-s + 2.55·19-s − 1.44·20-s + 2·22-s − 23-s − 2.89·25-s + 4.89·26-s − 28-s − 6.89·29-s + 6·31-s + 32-s + 2·34-s + 1.44·35-s + 11.7·37-s + 2.55·38-s − 1.44·40-s + 9.79·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.648·5-s − 0.377·7-s + 0.353·8-s − 0.458·10-s + 0.603·11-s + 1.35·13-s − 0.267·14-s + 0.250·16-s + 0.485·17-s + 0.585·19-s − 0.324·20-s + 0.426·22-s − 0.208·23-s − 0.579·25-s + 0.960·26-s − 0.188·28-s − 1.28·29-s + 1.07·31-s + 0.176·32-s + 0.342·34-s + 0.245·35-s + 1.93·37-s + 0.413·38-s − 0.229·40-s + 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.417177244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.417177244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 1.44T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2.55T + 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 - 6.89T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 - 6.55T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 0.101T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 2.89T + 97T^{2} \) |
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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
−9.793494786784438501371574931727, −9.046057312394664916492155775652, −7.944597885090500187252565392832, −7.37329037389303157850569248409, −6.16482514106445268229372327869, −5.79017344906158182702940994675, −4.30194463606659594268058444029, −3.81694937894669710576354089081, −2.78007502426693272047252555679, −1.15623785622867158010014689522,
1.15623785622867158010014689522, 2.78007502426693272047252555679, 3.81694937894669710576354089081, 4.30194463606659594268058444029, 5.79017344906158182702940994675, 6.16482514106445268229372327869, 7.37329037389303157850569248409, 7.944597885090500187252565392832, 9.046057312394664916492155775652, 9.793494786784438501371574931727
