| L(s) = 1 | − 2-s + 4-s + 3·5-s − 7-s − 8-s − 3·10-s + 5·13-s + 14-s + 16-s − 4·19-s + 3·20-s + 3·23-s + 4·25-s − 5·26-s − 28-s + 6·29-s − 2·31-s − 32-s − 3·35-s + 4·37-s + 4·38-s − 3·40-s + 6·41-s − 4·43-s − 3·46-s − 6·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s − 0.353·8-s − 0.948·10-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.670·20-s + 0.625·23-s + 4/5·25-s − 0.980·26-s − 0.188·28-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.507·35-s + 0.657·37-s + 0.648·38-s − 0.474·40-s + 0.937·41-s − 0.609·43-s − 0.442·46-s − 0.875·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.440351926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.440351926\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.96577903100166, −14.32361973377241, −13.82319386017309, −13.26390158188409, −12.95182299919543, −12.34730731008589, −11.60838434043018, −10.88841906210478, −10.73944148974996, −9.934847850008675, −9.712648489779042, −8.913484555907038, −8.691499817279514, −8.062236331332757, −7.257531955412200, −6.560937923645307, −6.237353478521880, −5.793385256990665, −5.051991255103727, −4.243261862818438, −3.440973465447747, −2.740158219878664, −2.087535881090385, −1.396304450810217, −0.6808857534930444,
0.6808857534930444, 1.396304450810217, 2.087535881090385, 2.740158219878664, 3.440973465447747, 4.243261862818438, 5.051991255103727, 5.793385256990665, 6.237353478521880, 6.560937923645307, 7.257531955412200, 8.062236331332757, 8.691499817279514, 8.913484555907038, 9.712648489779042, 9.934847850008675, 10.73944148974996, 10.88841906210478, 11.60838434043018, 12.34730731008589, 12.95182299919543, 13.26390158188409, 13.82319386017309, 14.32361973377241, 14.96577903100166
