| L(s) = 1 | − 3.00·2-s + 1.03·4-s − 10.0·5-s − 17.8·7-s + 20.9·8-s + 30.1·10-s + 34.7·11-s − 0.0479·13-s + 53.6·14-s − 71.2·16-s − 40.3·17-s − 119.·19-s − 10.3·20-s − 104.·22-s + 117.·23-s − 24.0·25-s + 0.144·26-s − 18.4·28-s − 46.1·29-s − 42.7·31-s + 46.4·32-s + 121.·34-s + 179.·35-s + 231.·37-s + 358.·38-s − 210.·40-s − 164.·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.129·4-s − 0.898·5-s − 0.964·7-s + 0.925·8-s + 0.954·10-s + 0.951·11-s − 0.00102·13-s + 1.02·14-s − 1.11·16-s − 0.575·17-s − 1.43·19-s − 0.116·20-s − 1.01·22-s + 1.06·23-s − 0.192·25-s + 0.00108·26-s − 0.124·28-s − 0.295·29-s − 0.247·31-s + 0.256·32-s + 0.611·34-s + 0.866·35-s + 1.03·37-s + 1.52·38-s − 0.831·40-s − 0.626·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3187518446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3187518446\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 3.00T + 8T^{2} \) |
| 5 | \( 1 + 10.0T + 125T^{2} \) |
| 7 | \( 1 + 17.8T + 343T^{2} \) |
| 11 | \( 1 - 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.0479T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 117.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 46.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 42.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 231.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 164.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 237.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 595.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 500.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 428.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 904.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 390.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 712.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 992.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 228.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.56e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 773.T + 9.12e5T^{2} \) |
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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
−8.760121354476479727159505373972, −8.217332675735978691651996876555, −7.11814034640339742058994101153, −6.83916973238085053917070586119, −5.73405217742528771836350853609, −4.31328437161399766095909210697, −4.01840667694745728889233184202, −2.75734414478968670108844847662, −1.46223692061642945224419549848, −0.31675860909299966425169552455,
0.31675860909299966425169552455, 1.46223692061642945224419549848, 2.75734414478968670108844847662, 4.01840667694745728889233184202, 4.31328437161399766095909210697, 5.73405217742528771836350853609, 6.83916973238085053917070586119, 7.11814034640339742058994101153, 8.217332675735978691651996876555, 8.760121354476479727159505373972
