| L(s) = 1 | + 2-s − 3-s + 4-s + 3.48·5-s − 6-s + 1.94·7-s + 8-s + 9-s + 3.48·10-s + 5.55·11-s − 12-s + 13-s + 1.94·14-s − 3.48·15-s + 16-s − 6.20·17-s + 18-s − 6.42·19-s + 3.48·20-s − 1.94·21-s + 5.55·22-s + 6.39·23-s − 24-s + 7.17·25-s + 26-s − 27-s + 1.94·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.56·5-s − 0.408·6-s + 0.736·7-s + 0.353·8-s + 0.333·9-s + 1.10·10-s + 1.67·11-s − 0.288·12-s + 0.277·13-s + 0.520·14-s − 0.900·15-s + 0.250·16-s − 1.50·17-s + 0.235·18-s − 1.47·19-s + 0.780·20-s − 0.425·21-s + 1.18·22-s + 1.33·23-s − 0.204·24-s + 1.43·25-s + 0.196·26-s − 0.192·27-s + 0.368·28-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
| \(L(1)\) |
\(\approx\) |
\(4.696978468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.696978468\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.48T + 5T^{2} \) |
| 7 | \( 1 - 1.94T + 7T^{2} \) |
| 11 | \( 1 - 5.55T + 11T^{2} \) |
| 17 | \( 1 + 6.20T + 17T^{2} \) |
| 19 | \( 1 + 6.42T + 19T^{2} \) |
| 23 | \( 1 - 6.39T + 23T^{2} \) |
| 29 | \( 1 + 3.68T + 29T^{2} \) |
| 31 | \( 1 + 6.42T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 4.25T + 43T^{2} \) |
| 47 | \( 1 + 3.04T + 47T^{2} \) |
| 53 | \( 1 + 3.06T + 53T^{2} \) |
| 59 | \( 1 + 4.29T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 6.04T + 71T^{2} \) |
| 73 | \( 1 - 1.77T + 73T^{2} \) |
| 79 | \( 1 + 3.88T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 0.650T + 89T^{2} \) |
| 97 | \( 1 - 6.72T + 97T^{2} \) |
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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
−7.58503852250350517655400240026, −6.70567291267973699606251508716, −6.31420472046654912598351812144, −5.91278804314735855420275892362, −4.98139796990093612192087842843, −4.46361986523925663753815027929, −3.76597923373863982907019835906, −2.40732059935600853203944030530, −1.88669017825097704106811181395, −1.07550989015818563090429926063,
1.07550989015818563090429926063, 1.88669017825097704106811181395, 2.40732059935600853203944030530, 3.76597923373863982907019835906, 4.46361986523925663753815027929, 4.98139796990093612192087842843, 5.91278804314735855420275892362, 6.31420472046654912598351812144, 6.70567291267973699606251508716, 7.58503852250350517655400240026
