L(s) = 1 | − 2-s + 0.539·3-s + 4-s − 3.38·5-s − 0.539·6-s − 8-s − 2.70·9-s + 3.38·10-s + 2.98·11-s + 0.539·12-s + 0.00964·13-s − 1.82·15-s + 16-s + 0.0684·17-s + 2.70·18-s − 0.966·19-s − 3.38·20-s − 2.98·22-s + 1.61·23-s − 0.539·24-s + 6.46·25-s − 0.00964·26-s − 3.08·27-s + 0.701·29-s + 1.82·30-s + 7.29·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.311·3-s + 0.5·4-s − 1.51·5-s − 0.220·6-s − 0.353·8-s − 0.902·9-s + 1.07·10-s + 0.900·11-s + 0.155·12-s + 0.00267·13-s − 0.471·15-s + 0.250·16-s + 0.0165·17-s + 0.638·18-s − 0.221·19-s − 0.757·20-s − 0.636·22-s + 0.337·23-s − 0.110·24-s + 1.29·25-s − 0.00189·26-s − 0.592·27-s + 0.130·29-s + 0.333·30-s + 1.30·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 0.539T + 3T^{2} \) |
| 5 | \( 1 + 3.38T + 5T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 - 0.00964T + 13T^{2} \) |
| 17 | \( 1 - 0.0684T + 17T^{2} \) |
| 19 | \( 1 + 0.966T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 - 0.701T + 29T^{2} \) |
| 31 | \( 1 - 7.29T + 31T^{2} \) |
| 37 | \( 1 - 4.88T + 37T^{2} \) |
| 43 | \( 1 - 1.84T + 43T^{2} \) |
| 47 | \( 1 + 3.16T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 + 3.16T + 59T^{2} \) |
| 61 | \( 1 + 8.67T + 61T^{2} \) |
| 67 | \( 1 - 8.55T + 67T^{2} \) |
| 71 | \( 1 + 3.64T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 6.62T + 83T^{2} \) |
| 89 | \( 1 + 9.28T + 89T^{2} \) |
| 97 | \( 1 - 4.16T + 97T^{2} \) |
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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
−8.009026681524729415501153903709, −7.72694051283163490708573172480, −6.71418070496032263424939582574, −6.17597611355614720845338517663, −4.95379274311440526932133971821, −4.08605742837137205019351567823, −3.34285543096341277714693329730, −2.56696916703774885288934006237, −1.14940014008191375502468423631, 0,
1.14940014008191375502468423631, 2.56696916703774885288934006237, 3.34285543096341277714693329730, 4.08605742837137205019351567823, 4.95379274311440526932133971821, 6.17597611355614720845338517663, 6.71418070496032263424939582574, 7.72694051283163490708573172480, 8.009026681524729415501153903709