L(s) = 1 | − i·2-s − i·3-s − 4-s + 2.66i·5-s − 6-s + i·8-s − 9-s + 2.66·10-s + (−2.85 − 1.68i)11-s + i·12-s + 2.50·13-s + 2.66·15-s + 16-s + 2.53·17-s + i·18-s − 1.39·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 1.19i·5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s + 0.843·10-s + (−0.860 − 0.509i)11-s + 0.288i·12-s + 0.693·13-s + 0.688·15-s + 0.250·16-s + 0.614·17-s + 0.235i·18-s − 0.319·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8227326660\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8227326660\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.85 + 1.68i)T \) |
good | 5 | \( 1 - 2.66iT - 5T^{2} \) |
| 13 | \( 1 - 2.50T + 13T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 23 | \( 1 + 3.15T + 23T^{2} \) |
| 29 | \( 1 - 6.51iT - 29T^{2} \) |
| 31 | \( 1 + 4.05iT - 31T^{2} \) |
| 37 | \( 1 - 2.54T + 37T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 - 5.62iT - 43T^{2} \) |
| 47 | \( 1 - 2.76iT - 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 + 4.91iT - 59T^{2} \) |
| 61 | \( 1 + 5.35T + 61T^{2} \) |
| 67 | \( 1 - 7.33T + 67T^{2} \) |
| 71 | \( 1 + 6.05T + 71T^{2} \) |
| 73 | \( 1 + 9.63T + 73T^{2} \) |
| 79 | \( 1 - 8.30iT - 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 11.2iT - 89T^{2} \) |
| 97 | \( 1 - 4.03iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.728620072423217627919584851775, −8.013479039487000550225995559652, −7.43943581272210691037431342522, −6.45611886113140262380950004040, −5.92460925818317484441575749654, −4.97364767194818675700170053270, −3.75068883459591743342757798234, −3.06848284946776147517760142874, −2.38748326763985511943083358372, −1.23889610445185734116132462880,
0.26507333614260582099960327447, 1.65998240541909535873736187158, 3.07526295607100153449673324124, 4.19700473325200427934412775520, 4.65511154610481882195914220859, 5.52250142631129695111949310962, 5.97510762618878629421787644804, 7.10968876319423416080249467340, 7.964248222443686315356542300922, 8.466943209393057744246287767287