| L(s) = 1 | + 2.34·2-s − 1.64·3-s + 3.51·4-s + 5-s − 3.86·6-s + 7-s + 3.56·8-s − 0.295·9-s + 2.34·10-s − 5.78·12-s − 0.774·13-s + 2.34·14-s − 1.64·15-s + 1.34·16-s − 4.83·17-s − 0.693·18-s + 6.49·19-s + 3.51·20-s − 1.64·21-s − 1.02·23-s − 5.86·24-s + 25-s − 1.82·26-s + 5.41·27-s + 3.51·28-s + 7.73·29-s − 3.86·30-s + ⋯ |
| L(s) = 1 | + 1.66·2-s − 0.949·3-s + 1.75·4-s + 0.447·5-s − 1.57·6-s + 0.377·7-s + 1.26·8-s − 0.0984·9-s + 0.742·10-s − 1.67·12-s − 0.214·13-s + 0.627·14-s − 0.424·15-s + 0.336·16-s − 1.17·17-s − 0.163·18-s + 1.48·19-s + 0.786·20-s − 0.358·21-s − 0.212·23-s − 1.19·24-s + 0.200·25-s − 0.357·26-s + 1.04·27-s + 0.665·28-s + 1.43·29-s − 0.705·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.149652087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.149652087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 3 | \( 1 + 1.64T + 3T^{2} \) |
| 13 | \( 1 + 0.774T + 13T^{2} \) |
| 17 | \( 1 + 4.83T + 17T^{2} \) |
| 19 | \( 1 - 6.49T + 19T^{2} \) |
| 23 | \( 1 + 1.02T + 23T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 - 9.88T + 31T^{2} \) |
| 37 | \( 1 - 1.89T + 37T^{2} \) |
| 41 | \( 1 - 9.56T + 41T^{2} \) |
| 43 | \( 1 + 2.28T + 43T^{2} \) |
| 47 | \( 1 - 2.03T + 47T^{2} \) |
| 53 | \( 1 - 4.32T + 53T^{2} \) |
| 59 | \( 1 - 1.32T + 59T^{2} \) |
| 61 | \( 1 + 0.894T + 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 1.66T + 73T^{2} \) |
| 79 | \( 1 + 1.15T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 6.45T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−8.232820967899290543988705152398, −7.25044407801717914415243497335, −6.45215615757216558872328772895, −6.10252604172595646124919461676, −5.26235530748536244178430073245, −4.82697890506762558765298391708, −4.16418510829709579248524653222, −2.96140721861984117322136389619, −2.38425180162185023622488146879, −0.960325056881719313960366564668,
0.960325056881719313960366564668, 2.38425180162185023622488146879, 2.96140721861984117322136389619, 4.16418510829709579248524653222, 4.82697890506762558765298391708, 5.26235530748536244178430073245, 6.10252604172595646124919461676, 6.45215615757216558872328772895, 7.25044407801717914415243497335, 8.232820967899290543988705152398
