| L(s) = 1 | − 2-s + 2.95·3-s + 4-s + 5-s − 2.95·6-s − 4.17·7-s − 8-s + 5.74·9-s − 10-s + 0.336·11-s + 2.95·12-s + 3.38·13-s + 4.17·14-s + 2.95·15-s + 16-s − 4.39·17-s − 5.74·18-s + 4.82·19-s + 20-s − 12.3·21-s − 0.336·22-s − 2.95·24-s + 25-s − 3.38·26-s + 8.11·27-s − 4.17·28-s + 7.34·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.70·3-s + 0.5·4-s + 0.447·5-s − 1.20·6-s − 1.57·7-s − 0.353·8-s + 1.91·9-s − 0.316·10-s + 0.101·11-s + 0.853·12-s + 0.938·13-s + 1.11·14-s + 0.763·15-s + 0.250·16-s − 1.06·17-s − 1.35·18-s + 1.10·19-s + 0.223·20-s − 2.69·21-s − 0.0717·22-s − 0.603·24-s + 0.200·25-s − 0.663·26-s + 1.56·27-s − 0.789·28-s + 1.36·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.650263849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.650263849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.95T + 3T^{2} \) |
| 7 | \( 1 + 4.17T + 7T^{2} \) |
| 11 | \( 1 - 0.336T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 29 | \( 1 - 7.34T + 29T^{2} \) |
| 31 | \( 1 + 5.33T + 31T^{2} \) |
| 37 | \( 1 + 8.12T + 37T^{2} \) |
| 41 | \( 1 + 0.0361T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 2.78T + 47T^{2} \) |
| 53 | \( 1 - 0.103T + 53T^{2} \) |
| 59 | \( 1 - 7.63T + 59T^{2} \) |
| 61 | \( 1 - 9.15T + 61T^{2} \) |
| 67 | \( 1 - 8.54T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 5.33T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 0.0786T + 83T^{2} \) |
| 89 | \( 1 - 9.18T + 89T^{2} \) |
| 97 | \( 1 - 3.60T + 97T^{2} \) |
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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
−8.502735717896084449436291440879, −7.55896858060833254200258113545, −6.90189985816030555786749685316, −6.41955542639111210847838493423, −5.45018379556622785608758945733, −4.03720569649258409984726594322, −3.45660242329783862654965889540, −2.76116502410735949637951280238, −2.08273672395177252282492213134, −0.905892713120333616056487201753,
0.905892713120333616056487201753, 2.08273672395177252282492213134, 2.76116502410735949637951280238, 3.45660242329783862654965889540, 4.03720569649258409984726594322, 5.45018379556622785608758945733, 6.41955542639111210847838493423, 6.90189985816030555786749685316, 7.55896858060833254200258113545, 8.502735717896084449436291440879
