L(s) = 1 | + (−1.41 − 1.73i)5-s + 2.82i·7-s − 4.89·11-s − 4.89i·13-s + 3.46i·17-s + 6.92·19-s + 4i·23-s + (−0.999 + 4.89i)25-s + 8.48·29-s + 6.92·31-s + (4.89 − 4.00i)35-s + 4.89i·37-s − 5.65·41-s + 11.3i·43-s − 4i·47-s + ⋯ |
L(s) = 1 | + (−0.632 − 0.774i)5-s + 1.06i·7-s − 1.47·11-s − 1.35i·13-s + 0.840i·17-s + 1.58·19-s + 0.834i·23-s + (−0.199 + 0.979i)25-s + 1.57·29-s + 1.24·31-s + (0.828 − 0.676i)35-s + 0.805i·37-s − 0.883·41-s + 1.72i·43-s − 0.583i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.230804244\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230804244\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.41 + 1.73i)T \) |
good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 11.3iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 9.79iT - 97T^{2} \) |
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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
−9.726763171973623825941782945922, −8.453026838915657132272134895495, −8.217592572752129456792195990013, −7.50710423423217914278777977778, −6.13433949214940589423952028083, −5.25311216590510760011827317587, −4.92623937438994587096466717060, −3.38465351393997329561772466976, −2.67753335213580441856715098047, −1.03727686841702577926665278090,
0.61710910129298858356457792409, 2.44615266102313414727694626372, 3.30772906145806578654120745264, 4.37027509543722738856199325097, 5.06635177922668515358240308913, 6.43502391573128965809123158827, 7.14703977158602591133418735600, 7.62889575811188101998331666727, 8.491541527157070356657306526072, 9.616720497759029784844867384641