| L(s) = 1 | + 2.02·2-s − 0.200·3-s + 2.08·4-s − 1.28·5-s − 0.405·6-s + 0.549·7-s + 0.163·8-s − 2.95·9-s − 2.60·10-s + 3.56·11-s − 0.417·12-s − 5.49·13-s + 1.11·14-s + 0.258·15-s − 3.83·16-s − 0.626·17-s − 5.97·18-s − 7.30·19-s − 2.67·20-s − 0.110·21-s + 7.20·22-s + 4.76·23-s − 0.0327·24-s − 3.34·25-s − 11.1·26-s + 1.19·27-s + 1.14·28-s + ⋯ |
| L(s) = 1 | + 1.42·2-s − 0.115·3-s + 1.04·4-s − 0.575·5-s − 0.165·6-s + 0.207·7-s + 0.0576·8-s − 0.986·9-s − 0.822·10-s + 1.07·11-s − 0.120·12-s − 1.52·13-s + 0.296·14-s + 0.0667·15-s − 0.958·16-s − 0.151·17-s − 1.40·18-s − 1.67·19-s − 0.599·20-s − 0.0240·21-s + 1.53·22-s + 0.994·23-s − 0.00668·24-s − 0.668·25-s − 2.17·26-s + 0.230·27-s + 0.216·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 + 0.200T + 3T^{2} \) |
| 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 - 0.549T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 + 0.626T + 17T^{2} \) |
| 19 | \( 1 + 7.30T + 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 - 4.81T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 - 2.95T + 37T^{2} \) |
| 41 | \( 1 + 3.93T + 41T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 + 9.94T + 53T^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 + 5.31T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 2.90T + 79T^{2} \) |
| 83 | \( 1 + 4.69T + 83T^{2} \) |
| 89 | \( 1 - 9.56T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−8.839793875074547102728035512148, −7.971302675999197005949403359070, −6.90566413717448645354081509564, −6.36864300136033387005090082259, −5.40566681321369546958052956746, −4.67469912270614403166309174668, −4.01292481773171379947453761438, −3.06562686496629541942228906578, −2.13802085959144756689095445531, 0,
2.13802085959144756689095445531, 3.06562686496629541942228906578, 4.01292481773171379947453761438, 4.67469912270614403166309174668, 5.40566681321369546958052956746, 6.36864300136033387005090082259, 6.90566413717448645354081509564, 7.971302675999197005949403359070, 8.839793875074547102728035512148
