L(s) = 1 | − 1.27i·2-s + 0.440i·3-s + 0.364·4-s + (−2.12 + 0.700i)5-s + 0.563·6-s + 2.70i·7-s − 3.02i·8-s + 2.80·9-s + (0.895 + 2.71i)10-s + 1.02·11-s + 0.160i·12-s + 1.80i·13-s + 3.45·14-s + (−0.308 − 0.936i)15-s − 3.13·16-s + 7.21i·17-s + ⋯ |
L(s) = 1 | − 0.904i·2-s + 0.254i·3-s + 0.182·4-s + (−0.949 + 0.313i)5-s + 0.230·6-s + 1.02i·7-s − 1.06i·8-s + 0.935·9-s + (0.283 + 0.858i)10-s + 0.308·11-s + 0.0464i·12-s + 0.501i·13-s + 0.923·14-s + (−0.0797 − 0.241i)15-s − 0.784·16-s + 1.75i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44198 + 0.231728i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44198 + 0.231728i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.12 - 0.700i)T \) |
| 151 | \( 1 + T \) |
good | 2 | \( 1 + 1.27iT - 2T^{2} \) |
| 3 | \( 1 - 0.440iT - 3T^{2} \) |
| 7 | \( 1 - 2.70iT - 7T^{2} \) |
| 11 | \( 1 - 1.02T + 11T^{2} \) |
| 13 | \( 1 - 1.80iT - 13T^{2} \) |
| 17 | \( 1 - 7.21iT - 17T^{2} \) |
| 19 | \( 1 + 8.01T + 19T^{2} \) |
| 23 | \( 1 - 3.91iT - 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 - 7.15T + 31T^{2} \) |
| 37 | \( 1 - 3.65iT - 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 9.33iT - 43T^{2} \) |
| 47 | \( 1 + 9.18iT - 47T^{2} \) |
| 53 | \( 1 + 6.89iT - 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 2.24iT - 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 11.9iT - 73T^{2} \) |
| 79 | \( 1 + 9.74T + 79T^{2} \) |
| 83 | \( 1 + 12.0iT - 83T^{2} \) |
| 89 | \( 1 + 3.57T + 89T^{2} \) |
| 97 | \( 1 - 5.42iT - 97T^{2} \) |
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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
−10.53941534147591498189778758401, −9.818810229763688740623320663264, −8.751702227372990764061702413046, −7.983916748754287539035536041071, −6.74070660129200447187294828683, −6.20404843409988972861324393509, −4.41460755327089136656896505628, −3.91720352149937693182806679897, −2.69688999130963081733730457618, −1.57739215114183848489914084467,
0.78849794286606774458733613670, 2.63314561244165934674194850416, 4.24317622185799118291136662006, 4.71331549652714295994576311081, 6.23743965956751545898471001074, 6.98471687967450388292186631905, 7.53965643566602798611450429566, 8.213857473440870906043538991805, 9.207361677306804195571402934014, 10.52938757627726988121261295991