| L(s) = 1 | + 2.78·2-s − 3-s + 5.73·4-s − 0.739·5-s − 2.78·6-s + 10.3·8-s + 9-s − 2.05·10-s + 4.44·11-s − 5.73·12-s − 0.996·13-s + 0.739·15-s + 17.4·16-s + 1.26·17-s + 2.78·18-s + 1.98·19-s − 4.24·20-s + 12.3·22-s + 0.818·23-s − 10.3·24-s − 4.45·25-s − 2.77·26-s − 27-s − 0.261·29-s + 2.05·30-s + 7.23·31-s + 27.6·32-s + ⋯ |
| L(s) = 1 | + 1.96·2-s − 0.577·3-s + 2.86·4-s − 0.330·5-s − 1.13·6-s + 3.67·8-s + 0.333·9-s − 0.650·10-s + 1.33·11-s − 1.65·12-s − 0.276·13-s + 0.190·15-s + 4.35·16-s + 0.306·17-s + 0.655·18-s + 0.454·19-s − 0.948·20-s + 2.63·22-s + 0.170·23-s − 2.12·24-s − 0.890·25-s − 0.543·26-s − 0.192·27-s − 0.0485·29-s + 0.375·30-s + 1.29·31-s + 4.89·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.068978950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.068978950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.78T + 2T^{2} \) |
| 5 | \( 1 + 0.739T + 5T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 + 0.996T + 13T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 - 0.818T + 23T^{2} \) |
| 29 | \( 1 + 0.261T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 + 7.70T + 37T^{2} \) |
| 43 | \( 1 + 4.72T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 7.87T + 59T^{2} \) |
| 61 | \( 1 + 6.85T + 61T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 + 5.60T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 1.81T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 7.76T + 97T^{2} \) |
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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
−7.53838567895261593153834418784, −7.12884796453041388544886857534, −6.30539489787543668420464389942, −5.95492960170484108456202274905, −5.04033407066057212236512969474, −4.52535159977197815661935834043, −3.76438383876618521482713210248, −3.22056578746422423050654406145, −2.08589634681692334158065167221, −1.16924115965295672259605224900,
1.16924115965295672259605224900, 2.08589634681692334158065167221, 3.22056578746422423050654406145, 3.76438383876618521482713210248, 4.52535159977197815661935834043, 5.04033407066057212236512969474, 5.95492960170484108456202274905, 6.30539489787543668420464389942, 7.12884796453041388544886857534, 7.53838567895261593153834418784
