| L(s) = 1 | − 3.16·3-s + 1.79·5-s + 6.99·9-s − 5.02·11-s + 13-s − 5.68·15-s − 1.34·17-s + 7.69·19-s + 0.0503·23-s − 1.76·25-s − 12.6·27-s − 0.101·29-s − 1.25·31-s + 15.8·33-s − 4.30·37-s − 3.16·39-s + 10.0·41-s + 3.21·43-s + 12.5·45-s − 9.76·47-s + 4.25·51-s + 2.99·53-s − 9.03·55-s − 24.3·57-s + 3.01·59-s + 12.1·61-s + 1.79·65-s + ⋯ |
| L(s) = 1 | − 1.82·3-s + 0.804·5-s + 2.33·9-s − 1.51·11-s + 0.277·13-s − 1.46·15-s − 0.326·17-s + 1.76·19-s + 0.0104·23-s − 0.353·25-s − 2.43·27-s − 0.0188·29-s − 0.224·31-s + 2.76·33-s − 0.708·37-s − 0.506·39-s + 1.56·41-s + 0.490·43-s + 1.87·45-s − 1.42·47-s + 0.595·51-s + 0.411·53-s − 1.21·55-s − 3.22·57-s + 0.392·59-s + 1.55·61-s + 0.223·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9452888582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9452888582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 - 1.79T + 5T^{2} \) |
| 11 | \( 1 + 5.02T + 11T^{2} \) |
| 17 | \( 1 + 1.34T + 17T^{2} \) |
| 19 | \( 1 - 7.69T + 19T^{2} \) |
| 23 | \( 1 - 0.0503T + 23T^{2} \) |
| 29 | \( 1 + 0.101T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 + 4.30T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 3.21T + 43T^{2} \) |
| 47 | \( 1 + 9.76T + 47T^{2} \) |
| 53 | \( 1 - 2.99T + 53T^{2} \) |
| 59 | \( 1 - 3.01T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 1.66T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 + 3.50T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.959284916495916329961381848242, −7.36030896406034452697941281123, −6.61966651909719841349629742659, −5.84083095209223084389947269241, −5.39089268070175687678974676440, −4.99500173385336318361553995461, −3.95261233382535620122191025115, −2.72769605723746517550073998539, −1.63725472621840736394481315754, −0.59494953041503438218341082120,
0.59494953041503438218341082120, 1.63725472621840736394481315754, 2.72769605723746517550073998539, 3.95261233382535620122191025115, 4.99500173385336318361553995461, 5.39089268070175687678974676440, 5.84083095209223084389947269241, 6.61966651909719841349629742659, 7.36030896406034452697941281123, 7.959284916495916329961381848242
