L(s) = 1 | − 2.61·2-s + 2.23·3-s + 4.85·4-s − 2.23·5-s − 5.85·6-s − 7.47·8-s + 2.00·9-s + 5.85·10-s − 3·11-s + 10.8·12-s + 13-s − 5.00·15-s + 9.85·16-s − 1.47·17-s − 5.23·18-s − 3·19-s − 10.8·20-s + 7.85·22-s − 8.23·23-s − 16.7·24-s − 2.61·26-s − 2.23·27-s + 4.47·29-s + 13.0·30-s − 5·31-s − 10.8·32-s − 6.70·33-s + ⋯ |
L(s) = 1 | − 1.85·2-s + 1.29·3-s + 2.42·4-s − 0.999·5-s − 2.38·6-s − 2.64·8-s + 0.666·9-s + 1.85·10-s − 0.904·11-s + 3.13·12-s + 0.277·13-s − 1.29·15-s + 2.46·16-s − 0.357·17-s − 1.23·18-s − 0.688·19-s − 2.42·20-s + 1.67·22-s − 1.71·23-s − 3.41·24-s − 0.513·26-s − 0.430·27-s + 0.830·29-s + 2.38·30-s − 0.898·31-s − 1.91·32-s − 1.16·33-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 + 7.47T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 + 3T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−9.926967043435727606751488411465, −9.073591398621732359811111073913, −8.279674204242944344175756530563, −7.979585550272571007696206837437, −7.27230104557089325010484293257, −6.07649062075562653452378841332, −4.09084762809047939404241567772, −2.90462871799594758035438569900, −1.95910020964794718294263088171, 0,
1.95910020964794718294263088171, 2.90462871799594758035438569900, 4.09084762809047939404241567772, 6.07649062075562653452378841332, 7.27230104557089325010484293257, 7.979585550272571007696206837437, 8.279674204242944344175756530563, 9.073591398621732359811111073913, 9.926967043435727606751488411465