L(s) = 1 | + (0.707 + 1.22i)2-s − 1.41·3-s + (−0.999 + 1.73i)4-s + (1.41 − 1.73i)5-s + (−1.00 − 1.73i)6-s − 2.44i·7-s − 2.82·8-s − 0.999·9-s + (3.12 + 0.507i)10-s + 3.46i·11-s + (1.41 − 2.44i)12-s + (2.99 − 1.73i)14-s + (−2.00 + 2.44i)15-s + (−2.00 − 3.46i)16-s + 4.89i·17-s + (−0.707 − 1.22i)18-s + ⋯ |
L(s) = 1 | + (0.499 + 0.866i)2-s − 0.816·3-s + (−0.499 + 0.866i)4-s + (0.632 − 0.774i)5-s + (−0.408 − 0.707i)6-s − 0.925i·7-s − 0.999·8-s − 0.333·9-s + (0.987 + 0.160i)10-s + 1.04i·11-s + (0.408 − 0.707i)12-s + (0.801 − 0.462i)14-s + (−0.516 + 0.632i)15-s + (−0.500 − 0.866i)16-s + 1.18i·17-s + (−0.166 − 0.288i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.707840 + 0.335869i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.707840 + 0.335869i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 2.44iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 + 7.34iT - 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−16.61869833173426828101495131259, −15.33402548569991036916178254446, −13.98146312766719566071658077906, −13.00226291846097029424403617274, −11.96711470515038797106701802629, −10.26052463905149426252027589742, −8.688515774099160318825793739805, −7.05017351554440068656211246655, −5.70799407082769302029455829041, −4.46317149075080133824051236363,
2.85942685608777736062149397746, 5.38378164260750889619052821507, 6.24336100117680943233153123675, 8.934312913185403525461999868856, 10.35701049142237880241621067005, 11.35752496417492728107494260346, 12.18709499912001420814971733755, 13.71670128727680470944692286223, 14.53651890546113625907385343274, 15.98129660576955663474552416353