| L(s) = 1 | + 1.70·3-s − 1.63·5-s + 0.460·7-s − 0.0783·9-s − 6.17·11-s − 4.07·13-s − 2.78·15-s + 0.630·17-s − 7.12·19-s + 0.787·21-s + 0.170·23-s − 2.34·25-s − 5.26·27-s + 2.63·29-s + 10.3·31-s − 10.5·33-s − 0.751·35-s + 5.12·37-s − 6.97·39-s + 3.15·41-s + 3.36·43-s + 0.127·45-s − 0.183·47-s − 6.78·49-s + 1.07·51-s − 4.34·53-s + 10.0·55-s + ⋯ |
| L(s) = 1 | + 0.986·3-s − 0.729·5-s + 0.174·7-s − 0.0261·9-s − 1.86·11-s − 1.13·13-s − 0.719·15-s + 0.153·17-s − 1.63·19-s + 0.171·21-s + 0.0354·23-s − 0.468·25-s − 1.01·27-s + 0.488·29-s + 1.86·31-s − 1.83·33-s − 0.127·35-s + 0.843·37-s − 1.11·39-s + 0.493·41-s + 0.513·43-s + 0.0190·45-s − 0.0267·47-s − 0.969·49-s + 0.151·51-s − 0.596·53-s + 1.35·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 3 | \( 1 - 1.70T + 3T^{2} \) |
| 5 | \( 1 + 1.63T + 5T^{2} \) |
| 7 | \( 1 - 0.460T + 7T^{2} \) |
| 11 | \( 1 + 6.17T + 11T^{2} \) |
| 13 | \( 1 + 4.07T + 13T^{2} \) |
| 17 | \( 1 - 0.630T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 0.170T + 23T^{2} \) |
| 29 | \( 1 - 2.63T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 3.15T + 41T^{2} \) |
| 43 | \( 1 - 3.36T + 43T^{2} \) |
| 47 | \( 1 + 0.183T + 47T^{2} \) |
| 53 | \( 1 + 4.34T + 53T^{2} \) |
| 59 | \( 1 + 1.78T + 59T^{2} \) |
| 67 | \( 1 - 8.55T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 0.552T + 73T^{2} \) |
| 79 | \( 1 - 7.00T + 79T^{2} \) |
| 83 | \( 1 + 6.83T + 83T^{2} \) |
| 89 | \( 1 - 4.49T + 89T^{2} \) |
| 97 | \( 1 - 8.52T + 97T^{2} \) |
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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
−9.595513463267748224006772812298, −8.428553271213384331372022819826, −8.033979652210388914996049874980, −7.47491044578941865481745698797, −6.19736124759227319591613490683, −4.99362966904548250960005766861, −4.20951341294398278993349588085, −2.87389729817823541904266662806, −2.36194406853278029740524105304, 0,
2.36194406853278029740524105304, 2.87389729817823541904266662806, 4.20951341294398278993349588085, 4.99362966904548250960005766861, 6.19736124759227319591613490683, 7.47491044578941865481745698797, 8.033979652210388914996049874980, 8.428553271213384331372022819826, 9.595513463267748224006772812298
