| L(s) = 1 | + 5.06·2-s + 17.6·4-s + 2.15·5-s + 2.11·7-s + 48.7·8-s + 10.9·10-s + 33.1·11-s + 6.68·13-s + 10.7·14-s + 105.·16-s − 55.3·17-s + 77.1·19-s + 38.0·20-s + 167.·22-s − 23·23-s − 120.·25-s + 33.8·26-s + 37.2·28-s − 14.9·29-s − 40.4·31-s + 146.·32-s − 280.·34-s + 4.55·35-s − 223.·37-s + 390.·38-s + 105.·40-s + 29.5·41-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 2.20·4-s + 0.192·5-s + 0.114·7-s + 2.15·8-s + 0.345·10-s + 0.908·11-s + 0.142·13-s + 0.204·14-s + 1.65·16-s − 0.789·17-s + 0.931·19-s + 0.425·20-s + 1.62·22-s − 0.208·23-s − 0.962·25-s + 0.255·26-s + 0.251·28-s − 0.0956·29-s − 0.234·31-s + 0.807·32-s − 1.41·34-s + 0.0220·35-s − 0.992·37-s + 1.66·38-s + 0.415·40-s + 0.112·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.229507399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.229507399\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 5.06T + 8T^{2} \) |
| 5 | \( 1 - 2.15T + 125T^{2} \) |
| 7 | \( 1 - 2.11T + 343T^{2} \) |
| 11 | \( 1 - 33.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 6.68T + 2.19e3T^{2} \) |
| 17 | \( 1 + 55.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 14.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 29.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 430.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 498.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 485.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 425.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 185.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 20.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 839.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 830.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 392.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 932.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 411.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.60e3T + 9.12e5T^{2} \) |
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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
−11.94957479153784496646848208992, −11.55073960858710209949796954104, −10.34292220396028127913267073183, −8.982750873550258390363408012058, −7.40450142604396386984681508169, −6.43566669833059959505477431725, −5.50319483670083435157510524295, −4.36834867687477614999862163444, −3.35754746972964286531860743478, −1.85991163886242491369445042400,
1.85991163886242491369445042400, 3.35754746972964286531860743478, 4.36834867687477614999862163444, 5.50319483670083435157510524295, 6.43566669833059959505477431725, 7.40450142604396386984681508169, 8.982750873550258390363408012058, 10.34292220396028127913267073183, 11.55073960858710209949796954104, 11.94957479153784496646848208992
