L(s) = 1 | + (0.192 + 1.40i)2-s + (−1.92 + 0.540i)4-s + 0.0802i·7-s + (−1.12 − 2.59i)8-s + 2.41i·11-s + 5.26·13-s + (−0.112 + 0.0154i)14-s + (3.41 − 2.08i)16-s + 0.255i·17-s − 6.95i·19-s + (−3.38 + 0.465i)22-s − 1.64i·23-s + (1.01 + 7.38i)26-s + (−0.0433 − 0.154i)28-s + 4.51i·29-s + ⋯ |
L(s) = 1 | + (0.136 + 0.990i)2-s + (−0.962 + 0.270i)4-s + 0.0303i·7-s + (−0.398 − 0.917i)8-s + 0.728i·11-s + 1.46·13-s + (−0.0300 + 0.00413i)14-s + (0.854 − 0.520i)16-s + 0.0620i·17-s − 1.59i·19-s + (−0.721 + 0.0993i)22-s − 0.343i·23-s + (0.199 + 1.44i)26-s + (−0.00819 − 0.0292i)28-s + 0.838i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0534 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0534 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.732160820\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.732160820\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.192 - 1.40i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.0802iT - 7T^{2} \) |
| 11 | \( 1 - 2.41iT - 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 - 0.255iT - 17T^{2} \) |
| 19 | \( 1 + 6.95iT - 19T^{2} \) |
| 23 | \( 1 + 1.64iT - 23T^{2} \) |
| 29 | \( 1 - 4.51iT - 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 + 4.08T + 43T^{2} \) |
| 47 | \( 1 + 5.70iT - 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 - 11.9iT - 61T^{2} \) |
| 67 | \( 1 - 7.27T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 5.50T + 79T^{2} \) |
| 83 | \( 1 - 9.20T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 8.50iT - 97T^{2} \) |
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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
−9.110260491643527002004439508187, −8.688399795072617351670841545349, −7.84369179253127366114951097752, −6.92142620305589639535761321865, −6.45235078406479573041725832925, −5.47657025023128544995402318545, −4.64963774662861535726847165598, −3.87983539133608682012995425726, −2.72402558799464435725660059253, −0.994835630014687299353259923872,
0.859864725954183693724207197434, 1.93306158027971086745720629381, 3.25795140786956993921843839185, 3.78262212866052142211292462225, 4.80334305701833238708268369172, 5.90603140355384127725380204517, 6.30532745964841989626320326164, 8.049843548164963710349359632013, 8.208049513795729438908050765780, 9.286448895511596004048183415048