| L(s) = 1 | − i·3-s − i·7-s + 2·9-s − 11-s + 6i·13-s + 7i·17-s − 19-s − 21-s + 8i·23-s − 5i·27-s + 6·29-s + 4·31-s + i·33-s − 8i·37-s + 6·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.377i·7-s + 0.666·9-s − 0.301·11-s + 1.66i·13-s + 1.69i·17-s − 0.229·19-s − 0.218·21-s + 1.66i·23-s − 0.962i·27-s + 1.11·29-s + 0.718·31-s + 0.174i·33-s − 1.31i·37-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.635165858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635165858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 - 15iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.721398022295779136169620725401, −8.700113316836445840078054960900, −7.981934832072708442320858249336, −7.04401845026294622597662082657, −6.60169324507038376327337793672, −5.58025139286356655472514970414, −4.34382941248300225089475117596, −3.76745244395628661583181959798, −2.15448702032286839235693112090, −1.33234153155859932189094019486,
0.73397297778883513839840362463, 2.55468970728621955049122464163, 3.27803713921887626994531108069, 4.72547677431999923314001115728, 4.98777396072363524285581432063, 6.20906592445487140484728905006, 7.05525234841462589816395295072, 8.044388465311646721934293537643, 8.646280040694369929067538312951, 9.745358688199461289818865521463
