L(s) = 1 | + 1.28·3-s − 3.57·5-s − 2.67·7-s − 1.34·9-s − 2.04·11-s − 0.441·13-s − 4.60·15-s − 5.96·17-s − 6.28·19-s − 3.44·21-s − 5.22·23-s + 7.75·25-s − 5.59·27-s + 9.50·29-s + 0.189·31-s − 2.63·33-s + 9.55·35-s − 9.10·37-s − 0.568·39-s + 7.36·41-s − 2.12·43-s + 4.78·45-s + 0.117·47-s + 0.159·49-s − 7.68·51-s − 7.37·53-s + 7.29·55-s + ⋯ |
L(s) = 1 | + 0.743·3-s − 1.59·5-s − 1.01·7-s − 0.446·9-s − 0.615·11-s − 0.122·13-s − 1.18·15-s − 1.44·17-s − 1.44·19-s − 0.752·21-s − 1.09·23-s + 1.55·25-s − 1.07·27-s + 1.76·29-s + 0.0340·31-s − 0.457·33-s + 1.61·35-s − 1.49·37-s − 0.0909·39-s + 1.15·41-s − 0.324·43-s + 0.713·45-s + 0.0171·47-s + 0.0227·49-s − 1.07·51-s − 1.01·53-s + 0.983·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2853107665\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2853107665\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 + 3.57T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 + 0.441T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 + 6.28T + 19T^{2} \) |
| 23 | \( 1 + 5.22T + 23T^{2} \) |
| 29 | \( 1 - 9.50T + 29T^{2} \) |
| 31 | \( 1 - 0.189T + 31T^{2} \) |
| 37 | \( 1 + 9.10T + 37T^{2} \) |
| 41 | \( 1 - 7.36T + 41T^{2} \) |
| 43 | \( 1 + 2.12T + 43T^{2} \) |
| 47 | \( 1 - 0.117T + 47T^{2} \) |
| 53 | \( 1 + 7.37T + 53T^{2} \) |
| 59 | \( 1 + 6.95T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 8.22T + 71T^{2} \) |
| 73 | \( 1 + 1.09T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 5.72T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 0.540T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.194462114237794295668285529211, −7.56491113045073970664487397364, −6.63941464303123843389946406004, −6.26958004776054962273418757562, −4.96482799873232795895719686234, −4.22298703990623541141075609960, −3.63412429315226152961502188084, −2.86480986066460812464642355583, −2.19531179735062428772493422961, −0.24693572420488352311102082588,
0.24693572420488352311102082588, 2.19531179735062428772493422961, 2.86480986066460812464642355583, 3.63412429315226152961502188084, 4.22298703990623541141075609960, 4.96482799873232795895719686234, 6.26958004776054962273418757562, 6.63941464303123843389946406004, 7.56491113045073970664487397364, 8.194462114237794295668285529211