| L(s) = 1 | + 2.56·2-s + 1.56·3-s + 4.56·4-s + 5-s + 4·6-s + 7-s + 6.56·8-s − 0.561·9-s + 2.56·10-s + 7.12·12-s − 0.438·13-s + 2.56·14-s + 1.56·15-s + 7.68·16-s + 0.438·17-s − 1.43·18-s + 7.12·19-s + 4.56·20-s + 1.56·21-s + 3.12·23-s + 10.2·24-s + 25-s − 1.12·26-s − 5.56·27-s + 4.56·28-s − 6.68·29-s + 4·30-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 0.901·3-s + 2.28·4-s + 0.447·5-s + 1.63·6-s + 0.377·7-s + 2.31·8-s − 0.187·9-s + 0.810·10-s + 2.05·12-s − 0.121·13-s + 0.684·14-s + 0.403·15-s + 1.92·16-s + 0.106·17-s − 0.339·18-s + 1.63·19-s + 1.01·20-s + 0.340·21-s + 0.651·23-s + 2.09·24-s + 0.200·25-s − 0.220·26-s − 1.07·27-s + 0.862·28-s − 1.24·29-s + 0.730·30-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
| \(L(1)\) |
\(\approx\) |
\(9.337144496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.337144496\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.56T + 2T^{2} \) |
| 3 | \( 1 - 1.56T + 3T^{2} \) |
| 13 | \( 1 + 0.438T + 13T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 + 0.876T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 + 5.12T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 2.43T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 - 5.80T + 97T^{2} \) |
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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
−8.076474363696145274866521427410, −7.57875859219602155072550103766, −6.76653129177464418176801902683, −5.93863727584563029726990417403, −5.27046477838754737738577509813, −4.74543113158459742090226715029, −3.61443848098398586797730894220, −3.18572319067969455754605102762, −2.38404939436388091178322230455, −1.52205660299310197849812197677,
1.52205660299310197849812197677, 2.38404939436388091178322230455, 3.18572319067969455754605102762, 3.61443848098398586797730894220, 4.74543113158459742090226715029, 5.27046477838754737738577509813, 5.93863727584563029726990417403, 6.76653129177464418176801902683, 7.57875859219602155072550103766, 8.076474363696145274866521427410
