| L(s) = 1 | − 0.566·3-s + 0.586·5-s − 1.33·7-s − 2.67·9-s − 5.79·11-s + 0.996·13-s − 0.332·15-s − 17-s − 5.01·19-s + 0.755·21-s − 5.70·23-s − 4.65·25-s + 3.21·27-s − 7.44·29-s + 5.98·31-s + 3.28·33-s − 0.782·35-s − 10.5·37-s − 0.564·39-s − 2.41·41-s + 8.43·43-s − 1.57·45-s + 7.75·47-s − 5.21·49-s + 0.566·51-s + 8.41·53-s − 3.40·55-s + ⋯ |
| L(s) = 1 | − 0.327·3-s + 0.262·5-s − 0.504·7-s − 0.893·9-s − 1.74·11-s + 0.276·13-s − 0.0857·15-s − 0.242·17-s − 1.14·19-s + 0.164·21-s − 1.18·23-s − 0.931·25-s + 0.619·27-s − 1.38·29-s + 1.07·31-s + 0.571·33-s − 0.132·35-s − 1.74·37-s − 0.0903·39-s − 0.376·41-s + 1.28·43-s − 0.234·45-s + 1.13·47-s − 0.745·49-s + 0.0793·51-s + 1.15·53-s − 0.458·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4201385374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4201385374\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 0.566T + 3T^{2} \) |
| 5 | \( 1 - 0.586T + 5T^{2} \) |
| 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 + 5.79T + 11T^{2} \) |
| 13 | \( 1 - 0.996T + 13T^{2} \) |
| 19 | \( 1 + 5.01T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 7.44T + 29T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 2.41T + 41T^{2} \) |
| 43 | \( 1 - 8.43T + 43T^{2} \) |
| 47 | \( 1 - 7.75T + 47T^{2} \) |
| 53 | \( 1 - 8.41T + 53T^{2} \) |
| 61 | \( 1 + 2.01T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 7.19T + 79T^{2} \) |
| 83 | \( 1 - 6.91T + 83T^{2} \) |
| 89 | \( 1 + 6.20T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.88127478659112085347392360249, −7.16103139327329515916617297167, −6.20609345622688154398507222697, −5.80741452635721633301688950664, −5.24963089637740484569945991352, −4.30685088132862654292102555625, −3.47445057419909216794351820936, −2.54757351694840360348801272785, −2.00714322682753295819440449265, −0.29954324189560638991019360018,
0.29954324189560638991019360018, 2.00714322682753295819440449265, 2.54757351694840360348801272785, 3.47445057419909216794351820936, 4.30685088132862654292102555625, 5.24963089637740484569945991352, 5.80741452635721633301688950664, 6.20609345622688154398507222697, 7.16103139327329515916617297167, 7.88127478659112085347392360249
