L(s) = 1 | − 2-s − 3-s + 4-s − 2.55·5-s + 6-s + 1.37·7-s − 8-s + 9-s + 2.55·10-s + 4.10·11-s − 12-s + 13-s − 1.37·14-s + 2.55·15-s + 16-s + 3.57·17-s − 18-s + 0.378·19-s − 2.55·20-s − 1.37·21-s − 4.10·22-s + 3.65·23-s + 24-s + 1.51·25-s − 26-s − 27-s + 1.37·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.14·5-s + 0.408·6-s + 0.520·7-s − 0.353·8-s + 0.333·9-s + 0.807·10-s + 1.23·11-s − 0.288·12-s + 0.277·13-s − 0.368·14-s + 0.659·15-s + 0.250·16-s + 0.866·17-s − 0.235·18-s + 0.0867·19-s − 0.570·20-s − 0.300·21-s − 0.875·22-s + 0.761·23-s + 0.204·24-s + 0.302·25-s − 0.196·26-s − 0.192·27-s + 0.260·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 2.55T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - 4.10T + 11T^{2} \) |
| 17 | \( 1 - 3.57T + 17T^{2} \) |
| 19 | \( 1 - 0.378T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 0.329T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 + 6.88T + 37T^{2} \) |
| 41 | \( 1 + 7.85T + 41T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 8.34T + 59T^{2} \) |
| 61 | \( 1 + 0.317T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 - 8.40T + 71T^{2} \) |
| 73 | \( 1 - 2.15T + 73T^{2} \) |
| 79 | \( 1 + 5.79T + 79T^{2} \) |
| 83 | \( 1 + 1.11T + 83T^{2} \) |
| 89 | \( 1 - 0.259T + 89T^{2} \) |
| 97 | \( 1 - 4.63T + 97T^{2} \) |
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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
−7.43494703036450142459728500695, −6.99020955473435697421695943657, −6.30289281436524659617134852263, −5.38153153970625845817177478398, −4.73062427364344171482356053231, −3.69467298131860357973060529329, −3.38756499587384812261273501490, −1.81844847898486280794005411894, −1.12598799712367628695757540994, 0,
1.12598799712367628695757540994, 1.81844847898486280794005411894, 3.38756499587384812261273501490, 3.69467298131860357973060529329, 4.73062427364344171482356053231, 5.38153153970625845817177478398, 6.30289281436524659617134852263, 6.99020955473435697421695943657, 7.43494703036450142459728500695