| L(s) = 1 | − 0.931·2-s + 0.912·3-s − 1.13·4-s − 2.28·5-s − 0.849·6-s + 7-s + 2.91·8-s − 2.16·9-s + 2.13·10-s + 5.30·11-s − 1.03·12-s + 6.15·13-s − 0.931·14-s − 2.08·15-s − 0.451·16-s + 5.13·17-s + 2.01·18-s − 4.92·19-s + 2.59·20-s + 0.912·21-s − 4.94·22-s + 8.31·23-s + 2.66·24-s + 0.237·25-s − 5.72·26-s − 4.71·27-s − 1.13·28-s + ⋯ |
| L(s) = 1 | − 0.658·2-s + 0.526·3-s − 0.566·4-s − 1.02·5-s − 0.346·6-s + 0.377·7-s + 1.03·8-s − 0.722·9-s + 0.674·10-s + 1.60·11-s − 0.298·12-s + 1.70·13-s − 0.248·14-s − 0.538·15-s − 0.112·16-s + 1.24·17-s + 0.475·18-s − 1.13·19-s + 0.579·20-s + 0.199·21-s − 1.05·22-s + 1.73·23-s + 0.543·24-s + 0.0475·25-s − 1.12·26-s − 0.907·27-s − 0.214·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9102905571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9102905571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 0.931T + 2T^{2} \) |
| 3 | \( 1 - 0.912T + 3T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 - 6.15T + 13T^{2} \) |
| 17 | \( 1 - 5.13T + 17T^{2} \) |
| 19 | \( 1 + 4.92T + 19T^{2} \) |
| 23 | \( 1 - 8.31T + 23T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 - 6.40T + 37T^{2} \) |
| 41 | \( 1 - 7.00T + 41T^{2} \) |
| 47 | \( 1 + 0.317T + 47T^{2} \) |
| 53 | \( 1 + 0.694T + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 + 1.21T + 61T^{2} \) |
| 67 | \( 1 - 3.09T + 67T^{2} \) |
| 71 | \( 1 + 5.16T + 71T^{2} \) |
| 73 | \( 1 + 1.48T + 73T^{2} \) |
| 79 | \( 1 + 9.61T + 79T^{2} \) |
| 83 | \( 1 + 4.35T + 83T^{2} \) |
| 89 | \( 1 + 3.50T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−11.35809464548087650191014869904, −11.02688305001809237413597140915, −9.494253524651106194756955262923, −8.711762142416300161861233480926, −8.298390527146297268704616113793, −7.25902463514678249016264780702, −5.82625702985494018522466802314, −4.20437956772191626578110282692, −3.52441995182725366633079679588, −1.18170761347905843553320325391,
1.18170761347905843553320325391, 3.52441995182725366633079679588, 4.20437956772191626578110282692, 5.82625702985494018522466802314, 7.25902463514678249016264780702, 8.298390527146297268704616113793, 8.711762142416300161861233480926, 9.494253524651106194756955262923, 11.02688305001809237413597140915, 11.35809464548087650191014869904
