| L(s) = 1 | − 1.68·2-s + 0.833·4-s − 4.59·7-s + 1.96·8-s − 3.91·11-s − 0.572·13-s + 7.72·14-s − 4.97·16-s + 0.232·17-s − 5.55·19-s + 6.59·22-s + 4.93·23-s + 0.964·26-s − 3.82·28-s − 4.13·29-s − 3.49·31-s + 4.44·32-s − 0.391·34-s − 5.41·37-s + 9.34·38-s − 10.4·41-s + 1.38·43-s − 3.26·44-s − 8.30·46-s − 0.920·47-s + 14.0·49-s − 0.477·52-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.416·4-s − 1.73·7-s + 0.694·8-s − 1.18·11-s − 0.158·13-s + 2.06·14-s − 1.24·16-s + 0.0564·17-s − 1.27·19-s + 1.40·22-s + 1.02·23-s + 0.189·26-s − 0.723·28-s − 0.767·29-s − 0.627·31-s + 0.785·32-s − 0.0671·34-s − 0.890·37-s + 1.51·38-s − 1.62·41-s + 0.211·43-s − 0.492·44-s − 1.22·46-s − 0.134·47-s + 2.01·49-s − 0.0662·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09021447394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09021447394\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.68T + 2T^{2} \) |
| 7 | \( 1 + 4.59T + 7T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 + 0.572T + 13T^{2} \) |
| 17 | \( 1 - 0.232T + 17T^{2} \) |
| 19 | \( 1 + 5.55T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 + 3.49T + 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 1.38T + 43T^{2} \) |
| 47 | \( 1 + 0.920T + 47T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 + 3.20T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 9.61T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 7.25T + 89T^{2} \) |
| 97 | \( 1 + 8.31T + 97T^{2} \) |
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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
−8.319267213969960098562235063166, −7.41020846210842739676364957307, −6.98924112613672358073797078738, −6.21450123782689574012003893622, −5.35672577939741259265319391571, −4.47705185079060277553823585315, −3.44849949508186702933564910674, −2.71561576686461094770881810993, −1.69029837900533550117149733739, −0.18502651377190533674831240743,
0.18502651377190533674831240743, 1.69029837900533550117149733739, 2.71561576686461094770881810993, 3.44849949508186702933564910674, 4.47705185079060277553823585315, 5.35672577939741259265319391571, 6.21450123782689574012003893622, 6.98924112613672358073797078738, 7.41020846210842739676364957307, 8.319267213969960098562235063166
