L(s) = 1 | + 2.08·2-s + 3.08i·3-s + 2.36·4-s + 1.51·5-s + 6.45i·6-s − i·7-s + 0.758·8-s − 6.54·9-s + 3.15·10-s − 5.15i·11-s + 7.30i·12-s − 1.61i·13-s − 2.08i·14-s + 4.67i·15-s − 3.14·16-s − 1.19i·17-s + ⋯ |
L(s) = 1 | + 1.47·2-s + 1.78i·3-s + 1.18·4-s + 0.676·5-s + 2.63i·6-s − 0.377i·7-s + 0.268·8-s − 2.18·9-s + 0.998·10-s − 1.55i·11-s + 2.10i·12-s − 0.446i·13-s − 0.558i·14-s + 1.20i·15-s − 0.785·16-s − 0.288i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.23202 + 1.56501i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23202 + 1.56501i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + iT \) |
| 41 | \( 1 + (6.01 + 2.18i)T \) |
good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 3 | \( 1 - 3.08iT - 3T^{2} \) |
| 5 | \( 1 - 1.51T + 5T^{2} \) |
| 11 | \( 1 + 5.15iT - 11T^{2} \) |
| 13 | \( 1 + 1.61iT - 13T^{2} \) |
| 17 | \( 1 + 1.19iT - 17T^{2} \) |
| 19 | \( 1 - 4.38iT - 19T^{2} \) |
| 23 | \( 1 - 7.15T + 23T^{2} \) |
| 29 | \( 1 - 7.89iT - 29T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 + 1.54T + 37T^{2} \) |
| 43 | \( 1 + 6.87T + 43T^{2} \) |
| 47 | \( 1 - 3.17iT - 47T^{2} \) |
| 53 | \( 1 + 4.55iT - 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 3.01iT - 67T^{2} \) |
| 71 | \( 1 + 14.5iT - 71T^{2} \) |
| 73 | \( 1 + 5.97T + 73T^{2} \) |
| 79 | \( 1 + 1.68iT - 79T^{2} \) |
| 83 | \( 1 - 5.09T + 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 6.18iT - 97T^{2} \) |
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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
−11.93241191313974698761577976554, −11.01146843688265281218788104343, −10.40073434080553831713393655608, −9.346361809948376556300450366844, −8.431094892955786603415074910655, −6.44204476015642690219400343506, −5.47390811338822254679282068308, −4.94980169550660403089786744309, −3.62486663589357107820045973822, −3.10230173338348164352001177088,
1.89120383038228931579923198955, 2.73258548446856405763507099172, 4.63871828906052336581990974295, 5.68361006526356660360168992319, 6.65258267867314025363693529356, 7.13068398950223825871759313320, 8.538404502884837876169014039066, 9.738587835004306170086083534951, 11.47803622220731833922063336822, 11.96357844359355898928082722917