L(s) = 1 | − 2.44·2-s − 0.667·3-s + 3.99·4-s + 0.910·5-s + 1.63·6-s − 4.87·8-s − 2.55·9-s − 2.22·10-s + 3.67·11-s − 2.66·12-s + 13-s − 0.607·15-s + 3.95·16-s + 7.18·17-s + 6.25·18-s − 1.97·19-s + 3.63·20-s − 9.00·22-s − 0.596·23-s + 3.25·24-s − 4.17·25-s − 2.44·26-s + 3.70·27-s − 3.64·29-s + 1.48·30-s − 7.08·31-s + 0.0786·32-s + ⋯ |
L(s) = 1 | − 1.73·2-s − 0.385·3-s + 1.99·4-s + 0.407·5-s + 0.667·6-s − 1.72·8-s − 0.851·9-s − 0.704·10-s + 1.10·11-s − 0.769·12-s + 0.277·13-s − 0.156·15-s + 0.987·16-s + 1.74·17-s + 1.47·18-s − 0.453·19-s + 0.812·20-s − 1.91·22-s − 0.124·23-s + 0.664·24-s − 0.834·25-s − 0.480·26-s + 0.713·27-s − 0.677·29-s + 0.271·30-s − 1.27·31-s + 0.0139·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6122053027\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6122053027\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 + 0.667T + 3T^{2} \) |
| 5 | \( 1 - 0.910T + 5T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 17 | \( 1 - 7.18T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 23 | \( 1 + 0.596T + 23T^{2} \) |
| 29 | \( 1 + 3.64T + 29T^{2} \) |
| 31 | \( 1 + 7.08T + 31T^{2} \) |
| 37 | \( 1 - 0.710T + 37T^{2} \) |
| 41 | \( 1 - 5.27T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 9.58T + 59T^{2} \) |
| 61 | \( 1 - 6.98T + 61T^{2} \) |
| 67 | \( 1 - 1.22T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 6.53T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 7.16T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 9.09T + 97T^{2} \) |
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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
−10.46466947777893245514847469291, −9.488190567588973886924029949260, −9.086066172778489900924812522211, −8.077294373094889802145067715754, −7.32571527757451021684283141226, −6.20714984492076393067546545162, −5.61686331131908083242876772755, −3.70465499603267629391003055171, −2.19542371399468063645972123031, −0.907931743152917920890736717421,
0.907931743152917920890736717421, 2.19542371399468063645972123031, 3.70465499603267629391003055171, 5.61686331131908083242876772755, 6.20714984492076393067546545162, 7.32571527757451021684283141226, 8.077294373094889802145067715754, 9.086066172778489900924812522211, 9.488190567588973886924029949260, 10.46466947777893245514847469291