| L(s) = 1 | − 2.39·2-s + 3-s + 3.73·4-s − 3.93·5-s − 2.39·6-s − 7-s − 4.14·8-s + 9-s + 9.42·10-s + 3.73·12-s − 2.99·13-s + 2.39·14-s − 3.93·15-s + 2.45·16-s − 6.60·17-s − 2.39·18-s − 5.90·19-s − 14.6·20-s − 21-s − 6.02·23-s − 4.14·24-s + 10.5·25-s + 7.17·26-s + 27-s − 3.73·28-s − 1.52·29-s + 9.42·30-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.577·3-s + 1.86·4-s − 1.76·5-s − 0.977·6-s − 0.377·7-s − 1.46·8-s + 0.333·9-s + 2.98·10-s + 1.07·12-s − 0.831·13-s + 0.639·14-s − 1.01·15-s + 0.614·16-s − 1.60·17-s − 0.564·18-s − 1.35·19-s − 3.28·20-s − 0.218·21-s − 1.25·23-s − 0.845·24-s + 2.10·25-s + 1.40·26-s + 0.192·27-s − 0.705·28-s − 0.282·29-s + 1.72·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2007897865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2007897865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 + 3.93T + 5T^{2} \) |
| 13 | \( 1 + 2.99T + 13T^{2} \) |
| 17 | \( 1 + 6.60T + 17T^{2} \) |
| 19 | \( 1 + 5.90T + 19T^{2} \) |
| 23 | \( 1 + 6.02T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 - 8.46T + 31T^{2} \) |
| 37 | \( 1 + 0.607T + 37T^{2} \) |
| 41 | \( 1 + 1.70T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 + 6.12T + 53T^{2} \) |
| 59 | \( 1 - 6.23T + 59T^{2} \) |
| 61 | \( 1 + 2.08T + 61T^{2} \) |
| 67 | \( 1 + 0.599T + 67T^{2} \) |
| 71 | \( 1 - 1.40T + 71T^{2} \) |
| 73 | \( 1 - 7.08T + 73T^{2} \) |
| 79 | \( 1 + 1.02T + 79T^{2} \) |
| 83 | \( 1 - 3.08T + 83T^{2} \) |
| 89 | \( 1 + 2.48T + 89T^{2} \) |
| 97 | \( 1 - 2.55T + 97T^{2} \) |
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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
−8.682277103243568131391943056176, −8.217828879915904793707489105046, −7.78938640094209403651839874849, −6.85031135682054897419808938758, −6.53074881967007815274320192259, −4.62343800008362918927658841033, −4.02512800180952902300076931925, −2.84674637708011702793350720291, −1.98846473844422060632767423234, −0.33320481613215275451844342677,
0.33320481613215275451844342677, 1.98846473844422060632767423234, 2.84674637708011702793350720291, 4.02512800180952902300076931925, 4.62343800008362918927658841033, 6.53074881967007815274320192259, 6.85031135682054897419808938758, 7.78938640094209403651839874849, 8.217828879915904793707489105046, 8.682277103243568131391943056176
