| L(s) = 1 | + 2-s − 2.89·3-s + 4-s − 2.40·5-s − 2.89·6-s + 7-s + 8-s + 5.40·9-s − 2.40·10-s + 5.55·11-s − 2.89·12-s + 4.10·13-s + 14-s + 6.95·15-s + 16-s + 4.58·17-s + 5.40·18-s − 1.32·19-s − 2.40·20-s − 2.89·21-s + 5.55·22-s + 3.75·23-s − 2.89·24-s + 0.763·25-s + 4.10·26-s − 6.95·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.67·3-s + 0.5·4-s − 1.07·5-s − 1.18·6-s + 0.377·7-s + 0.353·8-s + 1.80·9-s − 0.759·10-s + 1.67·11-s − 0.836·12-s + 1.13·13-s + 0.267·14-s + 1.79·15-s + 0.250·16-s + 1.11·17-s + 1.27·18-s − 0.303·19-s − 0.536·20-s − 0.632·21-s + 1.18·22-s + 0.782·23-s − 0.591·24-s + 0.152·25-s + 0.805·26-s − 1.33·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.043338214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.043338214\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 11 | \( 1 - 5.55T + 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 - 4.58T + 17T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 - 5.40T + 29T^{2} \) |
| 31 | \( 1 - 8.39T + 31T^{2} \) |
| 37 | \( 1 - 8.06T + 37T^{2} \) |
| 41 | \( 1 + 7.02T + 41T^{2} \) |
| 43 | \( 1 + 5.64T + 43T^{2} \) |
| 47 | \( 1 + 8.04T + 47T^{2} \) |
| 53 | \( 1 - 4.16T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 - 2.89T + 61T^{2} \) |
| 67 | \( 1 + 8.17T + 67T^{2} \) |
| 71 | \( 1 - 4.81T + 71T^{2} \) |
| 73 | \( 1 - 7.09T + 73T^{2} \) |
| 79 | \( 1 - 9.22T + 79T^{2} \) |
| 83 | \( 1 + 6.78T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.008501617357471258749386442215, −6.89425311152882638698813837434, −6.60214332758993146744932564359, −5.96649012794382022669053546592, −5.16666994550374899002108628341, −4.44381303163917477415645975201, −3.96941564658217645061750328711, −3.17185973088190480892110899567, −1.40117558222097970745926125825, −0.863755567354158697823475199392,
0.863755567354158697823475199392, 1.40117558222097970745926125825, 3.17185973088190480892110899567, 3.96941564658217645061750328711, 4.44381303163917477415645975201, 5.16666994550374899002108628341, 5.96649012794382022669053546592, 6.60214332758993146744932564359, 6.89425311152882638698813837434, 8.008501617357471258749386442215
