| L(s) = 1 | − 0.106·2-s + 3-s − 1.98·4-s − 1.41·5-s − 0.106·6-s + 7-s + 0.424·8-s + 9-s + 0.150·10-s − 3.00·11-s − 1.98·12-s − 0.106·14-s − 1.41·15-s + 3.93·16-s − 6.36·17-s − 0.106·18-s + 5.30·19-s + 2.80·20-s + 21-s + 0.319·22-s − 0.671·23-s + 0.424·24-s − 3.00·25-s + 27-s − 1.98·28-s + 0.516·29-s + 0.150·30-s + ⋯ |
| L(s) = 1 | − 0.0752·2-s + 0.577·3-s − 0.994·4-s − 0.631·5-s − 0.0434·6-s + 0.377·7-s + 0.149·8-s + 0.333·9-s + 0.0475·10-s − 0.904·11-s − 0.574·12-s − 0.0284·14-s − 0.364·15-s + 0.983·16-s − 1.54·17-s − 0.0250·18-s + 1.21·19-s + 0.628·20-s + 0.218·21-s + 0.0680·22-s − 0.140·23-s + 0.0865·24-s − 0.600·25-s + 0.192·27-s − 0.375·28-s + 0.0959·29-s + 0.0274·30-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
| \(L(1)\) |
\(\approx\) |
\(1.232603364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232603364\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.106T + 2T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 + 0.671T + 23T^{2} \) |
| 29 | \( 1 - 0.516T + 29T^{2} \) |
| 31 | \( 1 + 0.282T + 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 - 3.97T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 + 5.91T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.30T + 59T^{2} \) |
| 61 | \( 1 - 4.03T + 61T^{2} \) |
| 67 | \( 1 + 2.11T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 6.44T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 8.49T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 0.366T + 97T^{2} \) |
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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
−8.439602338945671457303324246115, −7.910391776797522435066156497512, −7.49050540866525391206605174763, −6.35802511610232548619549370983, −5.30117644640174139645729839749, −4.65133488501991561730266478759, −3.98572050801102141413350183935, −3.11827823976909757429644904986, −2.07307565418606907653518129469, −0.63729798751369223042645155824,
0.63729798751369223042645155824, 2.07307565418606907653518129469, 3.11827823976909757429644904986, 3.98572050801102141413350183935, 4.65133488501991561730266478759, 5.30117644640174139645729839749, 6.35802511610232548619549370983, 7.49050540866525391206605174763, 7.910391776797522435066156497512, 8.439602338945671457303324246115
