| L(s) = 1 | − 3.46·5-s − 24.2·7-s − 48·11-s + 41.5·13-s − 54·17-s − 4·19-s + 173.·23-s − 113·25-s − 162.·29-s + 58.8·31-s + 84·35-s − 325.·37-s − 294·41-s + 188·43-s − 505.·47-s + 245·49-s − 744.·53-s + 166.·55-s − 252·59-s − 90.0·61-s − 144·65-s + 628·67-s − 6.92·71-s + 1.00e3·73-s + 1.16e3·77-s − 1.34e3·79-s + 720·83-s + ⋯ |
| L(s) = 1 | − 0.309·5-s − 1.30·7-s − 1.31·11-s + 0.886·13-s − 0.770·17-s − 0.0482·19-s + 1.57·23-s − 0.904·25-s − 1.04·29-s + 0.341·31-s + 0.405·35-s − 1.44·37-s − 1.11·41-s + 0.666·43-s − 1.56·47-s + 0.714·49-s − 1.93·53-s + 0.407·55-s − 0.556·59-s − 0.189·61-s − 0.274·65-s + 1.14·67-s − 0.0115·71-s + 1.61·73-s + 1.72·77-s − 1.90·79-s + 0.952·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6465933606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6465933606\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.46T + 125T^{2} \) |
| 7 | \( 1 + 24.2T + 343T^{2} \) |
| 11 | \( 1 + 48T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 58.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 294T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188T + 7.95e4T^{2} \) |
| 47 | \( 1 + 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 744.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 252T + 2.05e5T^{2} \) |
| 61 | \( 1 + 90.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628T + 3.00e5T^{2} \) |
| 71 | \( 1 + 6.92T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 720T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−8.655430955944584362729689025672, −7.919340838308036289195149600747, −7.01805705152733811111974488598, −6.41693681530330771086557399053, −5.53526900193583728001193936892, −4.70375282267492794202573003259, −3.51566297451460690966381109263, −3.06549719186210529068662925749, −1.84857299912999640188911938285, −0.34231743596383099664392146484,
0.34231743596383099664392146484, 1.84857299912999640188911938285, 3.06549719186210529068662925749, 3.51566297451460690966381109263, 4.70375282267492794202573003259, 5.53526900193583728001193936892, 6.41693681530330771086557399053, 7.01805705152733811111974488598, 7.919340838308036289195149600747, 8.655430955944584362729689025672
