L(s) = 1 | + 2.16·2-s − 9.14i·3-s − 3.31·4-s − 16.2i·5-s − 19.7i·6-s + 12.2i·7-s − 24.4·8-s − 56.5·9-s − 35.1i·10-s − 11.1i·11-s + 30.2i·12-s + 28.8·13-s + 26.4i·14-s − 148.·15-s − 26.5·16-s + ⋯ |
L(s) = 1 | + 0.765·2-s − 1.75i·3-s − 0.414·4-s − 1.45i·5-s − 1.34i·6-s + 0.660i·7-s − 1.08·8-s − 2.09·9-s − 1.11i·10-s − 0.306i·11-s + 0.728i·12-s + 0.616·13-s + 0.505i·14-s − 2.55·15-s − 0.414·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.378961095\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.378961095\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - 2.16T + 8T^{2} \) |
| 3 | \( 1 + 9.14iT - 27T^{2} \) |
| 5 | \( 1 + 16.2iT - 125T^{2} \) |
| 7 | \( 1 - 12.2iT - 343T^{2} \) |
| 11 | \( 1 + 11.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 28.8T + 2.19e3T^{2} \) |
| 19 | \( 1 - 79.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.0iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 20.3iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 1.03iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 219. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 310. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 483.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 632.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 147.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 176. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 809.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 714. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 780. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 230. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 236.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 688.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.84e3iT - 9.12e5T^{2} \) |
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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
−11.50857598703378261830423380220, −9.444801678737598177923252246174, −8.550056475476659541305008069858, −8.105767562998550242903882052523, −6.56452977717525545928998296871, −5.67442642944717666334700563076, −4.90145269071228706585964032052, −3.21024472956880208296038514216, −1.59609083206107733879308540294, −0.41658086189744174866980937577,
3.12311894574710407737420309321, 3.61797148507357637736639647034, 4.63183875704019633513199397162, 5.63355801817640506876540145525, 6.77936882189359794895437429859, 8.266216015552184395679651798138, 9.603068755772873248942021155877, 9.981316241088659678142204031070, 11.03518553996724610392000358174, 11.53535099553979226936222553776