| L(s) = 1 | − 4.10·2-s + 8.83·4-s − 7.42·5-s + 5.67·7-s − 3.41·8-s + 30.4·10-s + 1.97·11-s + 0.731·13-s − 23.2·14-s − 56.6·16-s + 109.·17-s + 82.2·19-s − 65.5·20-s − 8.11·22-s + 189.·23-s − 69.9·25-s − 3.00·26-s + 50.1·28-s + 180.·29-s − 85.3·31-s + 259.·32-s − 449.·34-s − 42.1·35-s + 336.·37-s − 337.·38-s + 25.3·40-s − 337.·41-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 1.10·4-s − 0.663·5-s + 0.306·7-s − 0.151·8-s + 0.962·10-s + 0.0542·11-s + 0.0156·13-s − 0.444·14-s − 0.885·16-s + 1.56·17-s + 0.992·19-s − 0.732·20-s − 0.0786·22-s + 1.71·23-s − 0.559·25-s − 0.0226·26-s + 0.338·28-s + 1.15·29-s − 0.494·31-s + 1.43·32-s − 2.26·34-s − 0.203·35-s + 1.49·37-s − 1.43·38-s + 0.100·40-s − 1.28·41-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
| \(L(2)\) |
\(\approx\) |
\(1.111134592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.111134592\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.10T + 8T^{2} \) |
| 5 | \( 1 + 7.42T + 125T^{2} \) |
| 7 | \( 1 - 5.67T + 343T^{2} \) |
| 11 | \( 1 - 1.97T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.731T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 180.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 85.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 336.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 46.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 411.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 672.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 219.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 575.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 483.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 165.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 539.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 97.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 68.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + 96.0T + 9.12e5T^{2} \) |
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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
−8.710415413704288894809778542129, −7.975481668966004229581492317882, −7.49020238327023612837604573218, −6.84561262339680780893708732966, −5.61655270244623826447117072595, −4.75640481855578104685403025236, −3.63941768303076133261570887652, −2.65578522561883747401367160982, −1.27828713451049260190101299198, −0.70551837465678999613667549789,
0.70551837465678999613667549789, 1.27828713451049260190101299198, 2.65578522561883747401367160982, 3.63941768303076133261570887652, 4.75640481855578104685403025236, 5.61655270244623826447117072595, 6.84561262339680780893708732966, 7.49020238327023612837604573218, 7.975481668966004229581492317882, 8.710415413704288894809778542129
