L(s) = 1 | − 1.78i·3-s + (1.46 − 1.69i)5-s + 1.75i·7-s − 0.192·9-s − 4.77·11-s − 1.72i·13-s + (−3.02 − 2.61i)15-s − 7.81i·17-s + 2.43·19-s + 3.13·21-s + i·23-s + (−0.731 − 4.94i)25-s − 5.01i·27-s − 7.86·29-s + 6.14·31-s + ⋯ |
L(s) = 1 | − 1.03i·3-s + (0.653 − 0.757i)5-s + 0.663i·7-s − 0.0640·9-s − 1.43·11-s − 0.479i·13-s + (−0.780 − 0.673i)15-s − 1.89i·17-s + 0.559·19-s + 0.684·21-s + 0.208i·23-s + (−0.146 − 0.989i)25-s − 0.965i·27-s − 1.46·29-s + 1.10·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.611954 - 1.33641i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.611954 - 1.33641i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.46 + 1.69i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 + 1.78iT - 3T^{2} \) |
| 7 | \( 1 - 1.75iT - 7T^{2} \) |
| 11 | \( 1 + 4.77T + 11T^{2} \) |
| 13 | \( 1 + 1.72iT - 13T^{2} \) |
| 17 | \( 1 + 7.81iT - 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 - 6.14T + 31T^{2} \) |
| 37 | \( 1 + 6.83iT - 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 - 3.26iT - 43T^{2} \) |
| 47 | \( 1 - 8.46iT - 47T^{2} \) |
| 53 | \( 1 - 2.76iT - 53T^{2} \) |
| 59 | \( 1 + 1.91T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 + 12.7iT - 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 0.0111iT - 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 2.64iT - 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 15.8iT - 97T^{2} \) |
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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
−9.595190550696568845421774629930, −9.018786265847875297283125518843, −7.78026748421824141143834624659, −7.54347776741841818274737505809, −6.23869652047303499372452869136, −5.43248961304904939542159355541, −4.79554706358359470418346243442, −2.89021446239375467108641150774, −2.08413100349916062322362818818, −0.67982568968155642874172807630,
1.87356140930891123746607580634, 3.21569636895417067068233725685, 4.06138947718248227756667366633, 5.09365749652270624749351483968, 5.96465424770682619387971360247, 6.98535504871000996914388644397, 7.84243063616016471915740606826, 8.871964705089737847531596489496, 9.940417976822903033233289266845, 10.35212987194975860216403107416