L(s) = 1 | − 2.22·2-s − 3-s + 2.96·4-s + 4.15·5-s + 2.22·6-s − 2.15·8-s + 9-s − 9.25·10-s + 11-s − 2.96·12-s − 2.93·13-s − 4.15·15-s − 1.13·16-s + 1.36·17-s − 2.22·18-s + 4.10·19-s + 12.3·20-s − 2.22·22-s + 2.93·23-s + 2.15·24-s + 12.2·25-s + 6.53·26-s − 27-s − 8.97·29-s + 9.25·30-s + 4.10·31-s + 6.83·32-s + ⋯ |
L(s) = 1 | − 1.57·2-s − 0.577·3-s + 1.48·4-s + 1.85·5-s + 0.909·6-s − 0.760·8-s + 0.333·9-s − 2.92·10-s + 0.301·11-s − 0.856·12-s − 0.812·13-s − 1.07·15-s − 0.284·16-s + 0.331·17-s − 0.525·18-s + 0.941·19-s + 2.75·20-s − 0.475·22-s + 0.612·23-s + 0.439·24-s + 2.44·25-s + 1.28·26-s − 0.192·27-s − 1.66·29-s + 1.68·30-s + 0.737·31-s + 1.20·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9610492374\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9610492374\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 5 | \( 1 - 4.15T + 5T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 - 1.36T + 17T^{2} \) |
| 19 | \( 1 - 4.10T + 19T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 + 8.97T + 29T^{2} \) |
| 31 | \( 1 - 4.10T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 6.60T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 + 6.71T + 53T^{2} \) |
| 59 | \( 1 + 0.797T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 5.68T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 4.38T + 79T^{2} \) |
| 83 | \( 1 - 6.01T + 83T^{2} \) |
| 89 | \( 1 - 1.32T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−9.364454846431990368249901306548, −9.117448667808617407982780788616, −7.77268713137031912356768838131, −7.15464995512273697848814936674, −6.25866138484632604593617213660, −5.62362667671519481931827854912, −4.68659465608188143908176599984, −2.80676600450042750474140673821, −1.84786906912170371429727111946, −0.943943384607049759458929734095,
0.943943384607049759458929734095, 1.84786906912170371429727111946, 2.80676600450042750474140673821, 4.68659465608188143908176599984, 5.62362667671519481931827854912, 6.25866138484632604593617213660, 7.15464995512273697848814936674, 7.77268713137031912356768838131, 9.117448667808617407982780788616, 9.364454846431990368249901306548