L(s) = 1 | − 5.36i·2-s + 3·3-s − 20.7·4-s + 2.69i·5-s − 16.0i·6-s − 15.2i·7-s + 68.5i·8-s + 9·9-s + 14.4·10-s + 66.8i·11-s − 62.3·12-s − 81.5·14-s + 8.08i·15-s + 201.·16-s − 4.16·17-s − 48.2i·18-s + ⋯ |
L(s) = 1 | − 1.89i·2-s + 0.577·3-s − 2.59·4-s + 0.241i·5-s − 1.09i·6-s − 0.820i·7-s + 3.03i·8-s + 0.333·9-s + 0.457·10-s + 1.83i·11-s − 1.49·12-s − 1.55·14-s + 0.139i·15-s + 3.14·16-s − 0.0594·17-s − 0.632i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.779301233\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.779301233\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 5.36iT - 8T^{2} \) |
| 5 | \( 1 - 2.69iT - 125T^{2} \) |
| 7 | \( 1 + 15.2iT - 343T^{2} \) |
| 11 | \( 1 - 66.8iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 4.16T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 47.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 257.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 206. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 175. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 156. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 51.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 354. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 10.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 445. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 119.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 22.4iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 285. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 740. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 547.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 603. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 215. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3iT - 9.12e5T^{2} \) |
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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
−10.27655950215760783931189841857, −9.851377980319362600876613948260, −8.929231775144125100509621446820, −7.910534439542251444440081075782, −6.85146606591474907923447057618, −4.85084498978691356338703211252, −4.25887931355187885836117765209, −3.18638593296499806850493771039, −2.17028460103603498174159030750, −1.12421531522171722546494375747,
0.62644478671839557856125442633, 3.01092996635922061527062661039, 4.29152034160732056777995640011, 5.40096951497741483980913711595, 6.08956076494911793647634696193, 6.98364237101895621326563094376, 8.218277350790285565368030941522, 8.531555600196641983302471449986, 9.167489552437399497865577944926, 10.27517528344271093476030685187