L(s) = 1 | + (−2.23 + 2.23i)5-s + 4.57i·7-s + (1.74 − 1.74i)11-s + (1 + i)13-s − 6.47·17-s + (−1.74 − 1.74i)19-s − 5.65i·23-s − 5.00i·25-s + (0.236 + 0.236i)29-s − 10.2·31-s + (−10.2 − 10.2i)35-s + (−1.47 + 1.47i)37-s − 6.47i·41-s + (7.40 − 7.40i)43-s − 13.9·49-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.999i)5-s + 1.72i·7-s + (0.527 − 0.527i)11-s + (0.277 + 0.277i)13-s − 1.56·17-s + (−0.401 − 0.401i)19-s − 1.17i·23-s − 1.00i·25-s + (0.0438 + 0.0438i)29-s − 1.83·31-s + (−1.72 − 1.72i)35-s + (−0.242 + 0.242i)37-s − 1.01i·41-s + (1.12 − 1.12i)43-s − 1.99·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4082206487\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4082206487\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.23 - 2.23i)T - 5iT^{2} \) |
| 7 | \( 1 - 4.57iT - 7T^{2} \) |
| 11 | \( 1 + (-1.74 + 1.74i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + (1.74 + 1.74i)T + 19iT^{2} \) |
| 23 | \( 1 + 5.65iT - 23T^{2} \) |
| 29 | \( 1 + (-0.236 - 0.236i)T + 29iT^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + (1.47 - 1.47i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.47iT - 41T^{2} \) |
| 43 | \( 1 + (-7.40 + 7.40i)T - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-3.76 + 3.76i)T - 53iT^{2} \) |
| 59 | \( 1 + (6.32 - 6.32i)T - 59iT^{2} \) |
| 61 | \( 1 + (-7.47 - 7.47i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.48 - 8.48i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.49iT - 71T^{2} \) |
| 73 | \( 1 + 14.9iT - 73T^{2} \) |
| 79 | \( 1 - 1.08T + 79T^{2} \) |
| 83 | \( 1 + (-1.74 - 1.74i)T + 83iT^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 - 4.94T + 97T^{2} \) |
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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
−8.434267742307160565882151348363, −7.30464376559091333973763671450, −6.75428916139807184049545085797, −6.10849604895255094656316211447, −5.33770659625526652589314035672, −4.26730725309844943071152136963, −3.62241612960573486832769852532, −2.61830281478256907262126090708, −2.10118089584115989723021360105, −0.13555447661746361159902003890,
0.917105858888031127374870371013, 1.82790746508723381767135996511, 3.46803603867532306035375158688, 4.09533957570201414666757501932, 4.42907103998599442546374410861, 5.33008823086772862681907683542, 6.50705000454215920621772961077, 7.10317021486436460606145592429, 7.79003489317554660333475822998, 8.248130209079953075709179960713