L(s) = 1 | + 2i·3-s − 4i·5-s − 7-s − 9-s − 2i·11-s − 4i·13-s + 8·15-s + 2·17-s − 6i·19-s − 2i·21-s − 11·25-s + 4i·27-s + 8i·29-s + 8·31-s + 4·33-s + ⋯ |
L(s) = 1 | + 1.15i·3-s − 1.78i·5-s − 0.377·7-s − 0.333·9-s − 0.603i·11-s − 1.10i·13-s + 2.06·15-s + 0.485·17-s − 1.37i·19-s − 0.436i·21-s − 2.20·25-s + 0.769i·27-s + 1.48i·29-s + 1.43·31-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18160 - 0.489436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18160 - 0.489436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 5 | \( 1 + 4iT - 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 - 4iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.79551282548855389627948135389, −10.00078034210602862699336737032, −9.078664817954983034676137389722, −8.697769018473295927210536655281, −7.52069612891295355221778290769, −5.84301046069638269995488748734, −5.07135260317979897949845342147, −4.33115154610660680330821545485, −3.12865799059035804272392854294, −0.848289605798021387670017183136,
1.83048307998533227420738297881, 2.88507441518943687830078845100, 4.17607887309210232108380887483, 6.22330473762396035709990404524, 6.47455121033329918206541337099, 7.44222462391731258193780523312, 8.033104611333762006890392439678, 9.785440064046677561822638182324, 10.14845594714283680709060130124, 11.46956331397052785848454212106