L(s) = 1 | + 2.67i·2-s + 0.481i·3-s − 5.15·4-s + (1.67 − 1.48i)5-s − 1.28·6-s + 0.806i·7-s − 8.44i·8-s + 2.76·9-s + (3.96 + 4.48i)10-s − 3.67·11-s − 2.48i·12-s + i·13-s − 2.15·14-s + (0.712 + 0.806i)15-s + 12.2·16-s − 1.35i·17-s + ⋯ |
L(s) = 1 | + 1.89i·2-s + 0.277i·3-s − 2.57·4-s + (0.749 − 0.662i)5-s − 0.525·6-s + 0.304i·7-s − 2.98i·8-s + 0.922·9-s + (1.25 + 1.41i)10-s − 1.10·11-s − 0.716i·12-s + 0.277i·13-s − 0.576·14-s + (0.184 + 0.208i)15-s + 3.06·16-s − 0.327i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.354218 + 0.786041i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.354218 + 0.786041i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.67 + 1.48i)T \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 - 2.67iT - 2T^{2} \) |
| 3 | \( 1 - 0.481iT - 3T^{2} \) |
| 7 | \( 1 - 0.806iT - 7T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 17 | \( 1 + 1.35iT - 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 + 6.48iT - 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 - 3.76iT - 37T^{2} \) |
| 41 | \( 1 + 8.31T + 41T^{2} \) |
| 43 | \( 1 - 6.79iT - 43T^{2} \) |
| 47 | \( 1 - 3.19iT - 47T^{2} \) |
| 53 | \( 1 + 5.73iT - 53T^{2} \) |
| 59 | \( 1 + 5.98T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 - 9.89iT - 67T^{2} \) |
| 71 | \( 1 - 8.56T + 71T^{2} \) |
| 73 | \( 1 - 11.7iT - 73T^{2} \) |
| 79 | \( 1 - 2.26T + 79T^{2} \) |
| 83 | \( 1 + 3.84iT - 83T^{2} \) |
| 89 | \( 1 + 2.77T + 89T^{2} \) |
| 97 | \( 1 + 1.87iT - 97T^{2} \) |
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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
−15.54673816780158929299622790499, −14.47206298657666482533181744103, −13.37102741820397249382349950608, −12.68895429672687877718789726047, −10.16229805151599716975830888844, −9.227957740993045543645525656790, −8.140655141354650636749691191532, −6.85064122668876594296469897895, −5.51425556819673294298540917976, −4.60448092783219497061505181542,
1.91909961512465625173433187138, 3.50995963329224430598839239799, 5.32702882842433263307798493824, 7.55140802681601215059430881173, 9.339820764192456405679431270287, 10.28363942215524618960099815232, 10.89762629119522298221460593506, 12.28868952321884398013474176993, 13.29439790080493060087555043216, 13.74481344790317902117727778451