L(s) = 1 | − 2-s − 3-s + 4-s − 2.72·5-s + 6-s − 8-s + 9-s + 2.72·10-s + 3.36·11-s − 12-s − 6.56·13-s + 2.72·15-s + 16-s − 0.683·17-s − 18-s + 19-s − 2.72·20-s − 3.36·22-s + 1.07·23-s + 24-s + 2.40·25-s + 6.56·26-s − 27-s − 1.51·29-s − 2.72·30-s + 0.107·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.21·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.860·10-s + 1.01·11-s − 0.288·12-s − 1.82·13-s + 0.702·15-s + 0.250·16-s − 0.165·17-s − 0.235·18-s + 0.229·19-s − 0.608·20-s − 0.717·22-s + 0.224·23-s + 0.204·24-s + 0.481·25-s + 1.28·26-s − 0.192·27-s − 0.280·29-s − 0.496·30-s + 0.0192·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4584871365\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4584871365\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2.72T + 5T^{2} \) |
| 11 | \( 1 - 3.36T + 11T^{2} \) |
| 13 | \( 1 + 6.56T + 13T^{2} \) |
| 17 | \( 1 + 0.683T + 17T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 + 1.51T + 29T^{2} \) |
| 31 | \( 1 - 0.107T + 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 + 8.70T + 47T^{2} \) |
| 53 | \( 1 + 6.07T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 7.84T + 61T^{2} \) |
| 67 | \( 1 + 8.33T + 67T^{2} \) |
| 71 | \( 1 + 4.27T + 71T^{2} \) |
| 73 | \( 1 - 8.19T + 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 + 1.95T + 83T^{2} \) |
| 89 | \( 1 - 7.93T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.898160250219170156883842188210, −7.58314314357487047572297207483, −6.84645452898415061572127101652, −6.28073761188773088014572858942, −5.14962064001771034721913691318, −4.53064786825550355599968527995, −3.70953001275283046813336243955, −2.76918094261223001464796394918, −1.61150140686520767782156114051, −0.41679327508158865646351975146,
0.41679327508158865646351975146, 1.61150140686520767782156114051, 2.76918094261223001464796394918, 3.70953001275283046813336243955, 4.53064786825550355599968527995, 5.14962064001771034721913691318, 6.28073761188773088014572858942, 6.84645452898415061572127101652, 7.58314314357487047572297207483, 7.898160250219170156883842188210