| L(s) = 1 | − 2.79·2-s + 3-s + 5.79·4-s + 5-s − 2.79·6-s + 7-s − 10.5·8-s + 9-s − 2.79·10-s + 5·11-s + 5.79·12-s + 4.58·13-s − 2.79·14-s + 15-s + 17.9·16-s − 3·17-s − 2.79·18-s − 5.58·19-s + 5.79·20-s + 21-s − 13.9·22-s − 4·23-s − 10.5·24-s + 25-s − 12.7·26-s + 27-s + 5.79·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 0.577·3-s + 2.89·4-s + 0.447·5-s − 1.13·6-s + 0.377·7-s − 3.74·8-s + 0.333·9-s − 0.882·10-s + 1.50·11-s + 1.67·12-s + 1.27·13-s − 0.746·14-s + 0.258·15-s + 4.48·16-s − 0.727·17-s − 0.657·18-s − 1.28·19-s + 1.29·20-s + 0.218·21-s − 2.97·22-s − 0.834·23-s − 2.16·24-s + 0.200·25-s − 2.50·26-s + 0.192·27-s + 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8899734799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8899734799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 5.58T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 9.16T + 41T^{2} \) |
| 43 | \( 1 + 0.417T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 9.58T + 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 0.417T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 1.58T + 79T^{2} \) |
| 83 | \( 1 + 2.41T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
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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
−10.86001620996616501621968960803, −10.04687628451247281168330568363, −9.030539804290225604796074870475, −8.734331902902214329755345371240, −7.86646895476236338825536403591, −6.61450362517407997973927150168, −6.20655110506864942808847846597, −3.86913648466154764146989572946, −2.29058204194812420170594646686, −1.30728726876964145531883898151,
1.30728726876964145531883898151, 2.29058204194812420170594646686, 3.86913648466154764146989572946, 6.20655110506864942808847846597, 6.61450362517407997973927150168, 7.86646895476236338825536403591, 8.734331902902214329755345371240, 9.030539804290225604796074870475, 10.04687628451247281168330568363, 10.86001620996616501621968960803
