| L(s) = 1 | + 2.87·3-s − 5-s − 3.10·7-s + 5.28·9-s + 1.10·11-s + 1.77·13-s − 2.87·15-s + 7.75·17-s − 19-s − 8.93·21-s + 6.65·23-s + 25-s + 6.57·27-s + 7.75·29-s − 6.57·31-s + 3.18·33-s + 3.10·35-s − 1.40·37-s + 5.10·39-s + 2.81·41-s − 3.10·43-s − 5.28·45-s − 1.46·47-s + 2.63·49-s + 22.3·51-s − 2.59·53-s − 1.10·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s − 0.447·5-s − 1.17·7-s + 1.76·9-s + 0.333·11-s + 0.491·13-s − 0.743·15-s + 1.88·17-s − 0.229·19-s − 1.95·21-s + 1.38·23-s + 0.200·25-s + 1.26·27-s + 1.44·29-s − 1.18·31-s + 0.553·33-s + 0.524·35-s − 0.231·37-s + 0.817·39-s + 0.440·41-s − 0.473·43-s − 0.787·45-s − 0.213·47-s + 0.377·49-s + 3.12·51-s − 0.356·53-s − 0.148·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.779784844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.779784844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 - 1.10T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 - 7.75T + 17T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 - 7.75T + 29T^{2} \) |
| 31 | \( 1 + 6.57T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 - 2.81T + 41T^{2} \) |
| 43 | \( 1 + 3.10T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 + 2.59T + 53T^{2} \) |
| 59 | \( 1 - 5.38T + 59T^{2} \) |
| 61 | \( 1 - 4.07T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 7.14T + 71T^{2} \) |
| 73 | \( 1 - 0.243T + 73T^{2} \) |
| 79 | \( 1 - 9.38T + 79T^{2} \) |
| 83 | \( 1 + 8.86T + 83T^{2} \) |
| 89 | \( 1 - 0.813T + 89T^{2} \) |
| 97 | \( 1 - 3.19T + 97T^{2} \) |
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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
−9.407241583936220674768792082420, −8.653196944894383332605867871942, −8.024821709808815563452398089641, −7.19731382307738949024515331697, −6.50879879594255589014961887892, −5.25075764706544746706626150008, −3.89867620458813527639279247162, −3.37589461550456262780437641586, −2.71631329295999927797377785765, −1.19671858130083446499615453632,
1.19671858130083446499615453632, 2.71631329295999927797377785765, 3.37589461550456262780437641586, 3.89867620458813527639279247162, 5.25075764706544746706626150008, 6.50879879594255589014961887892, 7.19731382307738949024515331697, 8.024821709808815563452398089641, 8.653196944894383332605867871942, 9.407241583936220674768792082420
