L(s) = 1 | + 1.38i·2-s − 2.17·3-s + 0.0839·4-s + 4.26i·5-s − 3.01i·6-s + 2.88i·8-s + 1.74·9-s − 5.90·10-s + 0.143·11-s − 0.182·12-s − 2.01·13-s − 9.30i·15-s − 3.82·16-s + 2.60i·17-s + 2.42i·18-s + (−3.40 + 2.71i)19-s + ⋯ |
L(s) = 1 | + 0.978i·2-s − 1.25·3-s + 0.0419·4-s + 1.90i·5-s − 1.23i·6-s + 1.01i·8-s + 0.582·9-s − 1.86·10-s + 0.0431·11-s − 0.0528·12-s − 0.559·13-s − 2.40i·15-s − 0.956·16-s + 0.631i·17-s + 0.570i·18-s + (−0.782 + 0.623i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.396065 - 0.650282i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.396065 - 0.650282i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 + (3.40 - 2.71i)T \) |
good | 2 | \( 1 - 1.38iT - 2T^{2} \) |
| 3 | \( 1 + 2.17T + 3T^{2} \) |
| 5 | \( 1 - 4.26iT - 5T^{2} \) |
| 11 | \( 1 - 0.143T + 11T^{2} \) |
| 13 | \( 1 + 2.01T + 13T^{2} \) |
| 17 | \( 1 - 2.60iT - 17T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 - 7.53iT - 29T^{2} \) |
| 31 | \( 1 - 5.35T + 31T^{2} \) |
| 37 | \( 1 + 9.99iT - 37T^{2} \) |
| 41 | \( 1 + 1.27T + 41T^{2} \) |
| 43 | \( 1 - 1.66T + 43T^{2} \) |
| 47 | \( 1 + 7.99iT - 47T^{2} \) |
| 53 | \( 1 + 5.72iT - 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 6.32iT - 61T^{2} \) |
| 67 | \( 1 - 5.56iT - 67T^{2} \) |
| 71 | \( 1 - 4.99iT - 71T^{2} \) |
| 73 | \( 1 + 3.74iT - 73T^{2} \) |
| 79 | \( 1 + 1.47iT - 79T^{2} \) |
| 83 | \( 1 + 2.80iT - 83T^{2} \) |
| 89 | \( 1 - 2.80T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−10.70726226888813576643571564495, −10.22387768505683550481142342729, −8.769590134612348665117329158747, −7.60578629561899230957794762186, −6.97631662380131353685102532760, −6.42873744046200093472105440772, −5.82865481829573021424016366667, −4.89681571275827904364757776118, −3.38104779101930789726865347326, −2.21283395657664593672595915093,
0.44125663999454646914150072141, 1.29539309276089863021878423773, 2.75883973819364428770038893370, 4.53731520365921241913525005318, 4.72543852376907915257222710166, 5.86574515683468142785168447890, 6.72762605109649725444594645062, 7.923629411237809213519009454165, 9.015020054925172655405646266649, 9.614283704846005201676965229560