| L(s) = 1 | + 2-s − 0.183·3-s + 4-s + 5-s − 0.183·6-s − 3.52·7-s + 8-s − 2.96·9-s + 10-s + 11-s − 0.183·12-s − 4.25·13-s − 3.52·14-s − 0.183·15-s + 16-s − 0.133·17-s − 2.96·18-s + 2.57·19-s + 20-s + 0.645·21-s + 22-s + 3.03·23-s − 0.183·24-s + 25-s − 4.25·26-s + 1.09·27-s − 3.52·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.105·3-s + 0.5·4-s + 0.447·5-s − 0.0749·6-s − 1.33·7-s + 0.353·8-s − 0.988·9-s + 0.316·10-s + 0.301·11-s − 0.0529·12-s − 1.18·13-s − 0.940·14-s − 0.0473·15-s + 0.250·16-s − 0.0323·17-s − 0.699·18-s + 0.591·19-s + 0.223·20-s + 0.140·21-s + 0.213·22-s + 0.632·23-s − 0.0374·24-s + 0.200·25-s − 0.835·26-s + 0.210·27-s − 0.665·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.301172115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.301172115\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 0.183T + 3T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 + 0.133T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 - 3.03T + 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 - 9.53T + 31T^{2} \) |
| 37 | \( 1 - 5.20T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 47 | \( 1 - 7.87T + 47T^{2} \) |
| 53 | \( 1 - 7.91T + 53T^{2} \) |
| 59 | \( 1 + 9.81T + 59T^{2} \) |
| 61 | \( 1 - 7.68T + 61T^{2} \) |
| 67 | \( 1 - 9.85T + 67T^{2} \) |
| 71 | \( 1 - 4.62T + 71T^{2} \) |
| 73 | \( 1 - 1.46T + 73T^{2} \) |
| 79 | \( 1 + 5.97T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 7.08T + 89T^{2} \) |
| 97 | \( 1 - 6.08T + 97T^{2} \) |
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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
−8.258994548450186820439149777743, −7.31487258585144111032436252995, −6.65276025664646018276504162296, −6.10691088014456702262721854512, −5.38797441547693764884017894608, −4.71588309917514514475454507337, −3.63465308584670089930640663405, −2.88709662693370866465760293956, −2.35427986886721095932272631837, −0.72801842429990836773701507593,
0.72801842429990836773701507593, 2.35427986886721095932272631837, 2.88709662693370866465760293956, 3.63465308584670089930640663405, 4.71588309917514514475454507337, 5.38797441547693764884017894608, 6.10691088014456702262721854512, 6.65276025664646018276504162296, 7.31487258585144111032436252995, 8.258994548450186820439149777743
