| L(s) = 1 | + 0.554·2-s + 3-s − 1.69·4-s − 1.35·5-s + 0.554·6-s − 7-s − 2.04·8-s + 9-s − 0.753·10-s − 1.13·11-s − 1.69·12-s − 0.554·14-s − 1.35·15-s + 2.24·16-s − 0.554·17-s + 0.554·18-s + 4.10·19-s + 2.29·20-s − 21-s − 0.631·22-s − 6.78·23-s − 2.04·24-s − 3.15·25-s + 27-s + 1.69·28-s − 9.03·29-s − 0.753·30-s + ⋯ |
| L(s) = 1 | + 0.392·2-s + 0.577·3-s − 0.846·4-s − 0.606·5-s + 0.226·6-s − 0.377·7-s − 0.724·8-s + 0.333·9-s − 0.238·10-s − 0.342·11-s − 0.488·12-s − 0.148·14-s − 0.350·15-s + 0.561·16-s − 0.134·17-s + 0.130·18-s + 0.942·19-s + 0.513·20-s − 0.218·21-s − 0.134·22-s − 1.41·23-s − 0.418·24-s − 0.631·25-s + 0.192·27-s + 0.319·28-s − 1.67·29-s − 0.137·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.495828940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.495828940\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.554T + 2T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 11 | \( 1 + 1.13T + 11T^{2} \) |
| 17 | \( 1 + 0.554T + 17T^{2} \) |
| 19 | \( 1 - 4.10T + 19T^{2} \) |
| 23 | \( 1 + 6.78T + 23T^{2} \) |
| 29 | \( 1 + 9.03T + 29T^{2} \) |
| 31 | \( 1 - 8.63T + 31T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 41 | \( 1 + 0.219T + 41T^{2} \) |
| 43 | \( 1 - 3.89T + 43T^{2} \) |
| 47 | \( 1 - 8.13T + 47T^{2} \) |
| 53 | \( 1 - 7.98T + 53T^{2} \) |
| 59 | \( 1 + 3.89T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 0.762T + 67T^{2} \) |
| 71 | \( 1 + 4.50T + 71T^{2} \) |
| 73 | \( 1 + 8.50T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 8.60T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−8.532827802364826490000748192183, −7.82313437769601849772346967341, −7.36545603820178930946365208868, −6.11733129237227737557778129627, −5.55558711158395088786061701099, −4.49152883569045330607311354591, −3.93270744149503001315572893974, −3.25675031304454986003822401558, −2.25046730533651990444818771017, −0.64735097104755291503496902789,
0.64735097104755291503496902789, 2.25046730533651990444818771017, 3.25675031304454986003822401558, 3.93270744149503001315572893974, 4.49152883569045330607311354591, 5.55558711158395088786061701099, 6.11733129237227737557778129627, 7.36545603820178930946365208868, 7.82313437769601849772346967341, 8.532827802364826490000748192183
