L(s) = 1 | − 2.09·5-s + i·7-s + 0.961i·11-s + 4.60i·13-s + 3.28i·17-s − 3.66·19-s + 2.40·23-s − 0.618·25-s − 9.31·29-s − 1.84i·31-s − 2.09i·35-s + 2.29i·37-s − 0.314i·41-s + 8.64·43-s + 2.34·47-s + ⋯ |
L(s) = 1 | − 0.936·5-s + 0.377i·7-s + 0.289i·11-s + 1.27i·13-s + 0.795i·17-s − 0.841·19-s + 0.501·23-s − 0.123·25-s − 1.73·29-s − 0.332i·31-s − 0.353i·35-s + 0.377i·37-s − 0.0491i·41-s + 1.31·43-s + 0.341·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05045854468\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05045854468\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 2.09T + 5T^{2} \) |
| 11 | \( 1 - 0.961iT - 11T^{2} \) |
| 13 | \( 1 - 4.60iT - 13T^{2} \) |
| 17 | \( 1 - 3.28iT - 17T^{2} \) |
| 19 | \( 1 + 3.66T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 + 9.31T + 29T^{2} \) |
| 31 | \( 1 + 1.84iT - 31T^{2} \) |
| 37 | \( 1 - 2.29iT - 37T^{2} \) |
| 41 | \( 1 + 0.314iT - 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 - 2.34T + 47T^{2} \) |
| 53 | \( 1 + 7.73T + 53T^{2} \) |
| 59 | \( 1 + 7.99iT - 59T^{2} \) |
| 61 | \( 1 - 10.6iT - 61T^{2} \) |
| 67 | \( 1 + 4.12T + 67T^{2} \) |
| 71 | \( 1 + 6.79T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 + 1.26iT - 79T^{2} \) |
| 83 | \( 1 - 2.99iT - 83T^{2} \) |
| 89 | \( 1 + 12.8iT - 89T^{2} \) |
| 97 | \( 1 - 4.98T + 97T^{2} \) |
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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
−7.74280938952273675694757483772, −7.25874262817603995743416596287, −6.41268430222454003623898022183, −5.80036490252801530133971081740, −4.75533577970390840817355011051, −4.12212175220871405955403408337, −3.57314619817223057054010462066, −2.37293023103861520232011772982, −1.60213959991849731375978425502, −0.01567717841978473363710566037,
0.890233237215982089063002958698, 2.27968491216751601530798235296, 3.26391853765694894273534718258, 3.81951495600073655318089110244, 4.65964494283675564962531982481, 5.43950671821804569219644569199, 6.16562875231066709764205002569, 7.13203695544260913028983407339, 7.64956095291694491639816464261, 8.110866495327526567762285970120