L(s) = 1 | + 3-s + 7-s + 9-s + 4.77·11-s + 5.77·13-s − 3.77·17-s + 21-s + 3·23-s + 27-s + 3·29-s − 3.77·31-s + 4.77·33-s + 0.772·37-s + 5.77·39-s + 5.77·41-s − 4.54·43-s + 49-s − 3.77·51-s − 3.77·53-s + 5.77·59-s − 1.77·61-s + 63-s − 1.22·67-s + 3·69-s + 1.22·71-s + 2·73-s + 4.77·77-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 0.333·9-s + 1.43·11-s + 1.60·13-s − 0.914·17-s + 0.218·21-s + 0.625·23-s + 0.192·27-s + 0.557·29-s − 0.677·31-s + 0.830·33-s + 0.126·37-s + 0.924·39-s + 0.901·41-s − 0.692·43-s + 0.142·49-s − 0.528·51-s − 0.518·53-s + 0.751·59-s − 0.226·61-s + 0.125·63-s − 0.150·67-s + 0.361·69-s + 0.145·71-s + 0.234·73-s + 0.543·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.083673071\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.083673071\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 - 5.77T + 13T^{2} \) |
| 17 | \( 1 + 3.77T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 3.77T + 31T^{2} \) |
| 37 | \( 1 - 0.772T + 37T^{2} \) |
| 41 | \( 1 - 5.77T + 41T^{2} \) |
| 43 | \( 1 + 4.54T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.77T + 53T^{2} \) |
| 59 | \( 1 - 5.77T + 59T^{2} \) |
| 61 | \( 1 + 1.77T + 61T^{2} \) |
| 67 | \( 1 + 1.22T + 67T^{2} \) |
| 71 | \( 1 - 1.22T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 0.455T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.564125929222245225438841281859, −7.80778728436110762376553665567, −6.79466853088417731867587328010, −6.43601585520083427473315652409, −5.47200357624678074070071315245, −4.37201146610065101268986073322, −3.88878696228427796038604877424, −3.02299978118055796504840715675, −1.84205877857434540832259520160, −1.07207460207290721352108095179,
1.07207460207290721352108095179, 1.84205877857434540832259520160, 3.02299978118055796504840715675, 3.88878696228427796038604877424, 4.37201146610065101268986073322, 5.47200357624678074070071315245, 6.43601585520083427473315652409, 6.79466853088417731867587328010, 7.80778728436110762376553665567, 8.564125929222245225438841281859