| L(s) = 1 | − 2.20·2-s − 3·3-s − 3.12·4-s + 6.62·6-s + 1.50·7-s + 24.5·8-s + 9·9-s + 11·11-s + 9.38·12-s − 68.3·13-s − 3.33·14-s − 29.1·16-s + 113.·17-s − 19.8·18-s − 72.3·19-s − 4.52·21-s − 24.2·22-s + 144.·23-s − 73.6·24-s + 150.·26-s − 27·27-s − 4.72·28-s − 133.·29-s + 177.·31-s − 132.·32-s − 33·33-s − 249.·34-s + ⋯ |
| L(s) = 1 | − 0.780·2-s − 0.577·3-s − 0.391·4-s + 0.450·6-s + 0.0815·7-s + 1.08·8-s + 0.333·9-s + 0.301·11-s + 0.225·12-s − 1.45·13-s − 0.0636·14-s − 0.455·16-s + 1.61·17-s − 0.260·18-s − 0.873·19-s − 0.0470·21-s − 0.235·22-s + 1.31·23-s − 0.626·24-s + 1.13·26-s − 0.192·27-s − 0.0318·28-s − 0.854·29-s + 1.03·31-s − 0.729·32-s − 0.174·33-s − 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7050050700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7050050700\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 2.20T + 8T^{2} \) |
| 7 | \( 1 - 1.50T + 343T^{2} \) |
| 13 | \( 1 + 68.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 113.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 72.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 144.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 39.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 366.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 427.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 340.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 659.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 525.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 462.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 514.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 848.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 987.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 442.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 468.T + 9.12e5T^{2} \) |
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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
−9.936044350975767509102128134930, −9.113608912799933005046300761995, −8.139591728668500457210634359667, −7.42309848808239754136706917963, −6.53759082138304742266947282509, −5.20085657486508927810564814911, −4.72059711785863941584099248002, −3.36166193263693169317819597330, −1.73722395275093869298364761848, −0.55694280389870574774357951206,
0.55694280389870574774357951206, 1.73722395275093869298364761848, 3.36166193263693169317819597330, 4.72059711785863941584099248002, 5.20085657486508927810564814911, 6.53759082138304742266947282509, 7.42309848808239754136706917963, 8.139591728668500457210634359667, 9.113608912799933005046300761995, 9.936044350975767509102128134930
