L(s) = 1 | + (1.16 + 0.796i)2-s + i·3-s + (0.731 + 1.86i)4-s + (−0.796 + 1.16i)6-s − 4.72·7-s + (−0.627 + 2.75i)8-s − 9-s + 3.93i·11-s + (−1.86 + 0.731i)12-s − 3.46i·13-s + (−5.51 − 3.76i)14-s + (−2.93 + 2.72i)16-s − 3.51·17-s + (−1.16 − 0.796i)18-s + 5.44i·19-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)2-s + 0.577i·3-s + (0.365 + 0.930i)4-s + (−0.325 + 0.477i)6-s − 1.78·7-s + (−0.221 + 0.975i)8-s − 0.333·9-s + 1.18i·11-s + (−0.537 + 0.211i)12-s − 0.961i·13-s + (−1.47 − 1.00i)14-s + (−0.732 + 0.680i)16-s − 0.852·17-s + (−0.275 − 0.187i)18-s + 1.24i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.170781 + 1.51964i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.170781 + 1.51964i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.16 - 0.796i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4.72T + 7T^{2} \) |
| 11 | \( 1 - 3.93iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 - 5.44iT - 19T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 - 3.66iT - 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 0.414iT - 37T^{2} \) |
| 41 | \( 1 - 3.00T + 41T^{2} \) |
| 43 | \( 1 - 5.34iT - 43T^{2} \) |
| 47 | \( 1 - 0.925T + 47T^{2} \) |
| 53 | \( 1 + 0.233iT - 53T^{2} \) |
| 59 | \( 1 + 14.3iT - 59T^{2} \) |
| 61 | \( 1 - 0.118iT - 61T^{2} \) |
| 67 | \( 1 - 13.4iT - 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 - 0.563T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 - 8.88T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−11.04966961338269131631307783024, −10.10957022301563639240170964702, −9.421114710892092884881375134428, −8.376240233902104599818220807796, −7.20327985062965083877440358310, −6.51274375105669008042307614112, −5.59420562641212824519884618071, −4.55716357765669925569954760438, −3.52475047017929445879681177405, −2.70544020546026022641777625600,
0.62280680327511820602013929160, 2.53586987360629543266000101036, 3.26713608800385983286257918733, 4.47252596822288312361998375702, 5.82146562749990109371727854027, 6.55737072265480307788124034358, 7.05852679154603381357474675503, 8.891721650704399877276085034859, 9.328083766898338176692344501953, 10.51140794190026057118534973507