| L(s) = 1 | − 2.12·3-s − 0.932·5-s − 0.358·7-s + 1.52·9-s − 11-s − 0.852·13-s + 1.98·15-s − 5.36·17-s − 6.45·19-s + 0.763·21-s − 4.52·23-s − 4.13·25-s + 3.13·27-s − 6.95·29-s + 3.01·31-s + 2.12·33-s + 0.334·35-s + 1.01·37-s + 1.81·39-s + 5.66·41-s − 6.95·43-s − 1.42·45-s + 5.16·47-s − 6.87·49-s + 11.4·51-s + 8.47·53-s + 0.932·55-s + ⋯ |
| L(s) = 1 | − 1.22·3-s − 0.416·5-s − 0.135·7-s + 0.508·9-s − 0.301·11-s − 0.236·13-s + 0.511·15-s − 1.30·17-s − 1.48·19-s + 0.166·21-s − 0.944·23-s − 0.826·25-s + 0.603·27-s − 1.29·29-s + 0.541·31-s + 0.370·33-s + 0.0565·35-s + 0.166·37-s + 0.290·39-s + 0.884·41-s − 1.06·43-s − 0.211·45-s + 0.752·47-s − 0.981·49-s + 1.59·51-s + 1.16·53-s + 0.125·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1750800275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1750800275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 + 2.12T + 3T^{2} \) |
| 5 | \( 1 + 0.932T + 5T^{2} \) |
| 7 | \( 1 + 0.358T + 7T^{2} \) |
| 13 | \( 1 + 0.852T + 13T^{2} \) |
| 17 | \( 1 + 5.36T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 29 | \( 1 + 6.95T + 29T^{2} \) |
| 31 | \( 1 - 3.01T + 31T^{2} \) |
| 37 | \( 1 - 1.01T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 + 6.95T + 43T^{2} \) |
| 47 | \( 1 - 5.16T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 7.89T + 59T^{2} \) |
| 61 | \( 1 + 2.89T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 + 5.50T + 71T^{2} \) |
| 73 | \( 1 + 9.53T + 73T^{2} \) |
| 79 | \( 1 - 8.14T + 79T^{2} \) |
| 83 | \( 1 + 3.87T + 83T^{2} \) |
| 89 | \( 1 + 2.84T + 89T^{2} \) |
| 97 | \( 1 + 5.54T + 97T^{2} \) |
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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
−8.049215439217250961834440390565, −7.24071506983098043963818903398, −6.48244905394727130285934848968, −6.02333500108306421504814343992, −5.29597872498377221813098859431, −4.39509998197006516977035771194, −4.00712001884882064287886472187, −2.67969324026844005667818250018, −1.79031823991712102806804307341, −0.22295317421649602038592441170,
0.22295317421649602038592441170, 1.79031823991712102806804307341, 2.67969324026844005667818250018, 4.00712001884882064287886472187, 4.39509998197006516977035771194, 5.29597872498377221813098859431, 6.02333500108306421504814343992, 6.48244905394727130285934848968, 7.24071506983098043963818903398, 8.049215439217250961834440390565
