| L(s) = 1 | + 3-s − 5-s + 1.35·7-s + 9-s + 3.58·11-s + 13-s − 15-s − 7.81·17-s + 4.94·19-s + 1.35·21-s + 0.0737·23-s + 25-s + 27-s + 5.51·29-s + 4·31-s + 3.58·33-s − 1.35·35-s − 1.58·37-s + 39-s − 5.58·41-s + 7.17·43-s − 45-s + 2.56·47-s − 5.15·49-s − 7.81·51-s + 12.7·53-s − 3.58·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.513·7-s + 0.333·9-s + 1.08·11-s + 0.277·13-s − 0.258·15-s − 1.89·17-s + 1.13·19-s + 0.296·21-s + 0.0153·23-s + 0.200·25-s + 0.192·27-s + 1.02·29-s + 0.718·31-s + 0.624·33-s − 0.229·35-s − 0.261·37-s + 0.160·39-s − 0.872·41-s + 1.09·43-s − 0.149·45-s + 0.374·47-s − 0.736·49-s − 1.09·51-s + 1.75·53-s − 0.483·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.708695771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.708695771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 17 | \( 1 + 7.81T + 17T^{2} \) |
| 19 | \( 1 - 4.94T + 19T^{2} \) |
| 23 | \( 1 - 0.0737T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 1.58T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 - 7.17T + 43T^{2} \) |
| 47 | \( 1 - 2.56T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 8.45T + 59T^{2} \) |
| 61 | \( 1 + 4.30T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 0.871T + 71T^{2} \) |
| 73 | \( 1 - 6.79T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 + 6.87T + 89T^{2} \) |
| 97 | \( 1 - 7.35T + 97T^{2} \) |
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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
−8.172404534655668676227531354208, −7.32458156123370363064492509504, −6.76564135494387278583369946505, −6.08221945284508279113644741136, −4.92471754280951581694322969201, −4.38110104469160382783253654372, −3.67741972315176807732138226514, −2.79660872319380591250899830837, −1.85472130370154996300596592429, −0.869081742809182179977270810884,
0.869081742809182179977270810884, 1.85472130370154996300596592429, 2.79660872319380591250899830837, 3.67741972315176807732138226514, 4.38110104469160382783253654372, 4.92471754280951581694322969201, 6.08221945284508279113644741136, 6.76564135494387278583369946505, 7.32458156123370363064492509504, 8.172404534655668676227531354208
