| L(s) = 1 | + 2-s + 3-s + 4-s − 1.81·5-s + 6-s + 4.37·7-s + 8-s + 9-s − 1.81·10-s − 5.56·11-s + 12-s − 13-s + 4.37·14-s − 1.81·15-s + 16-s − 5.16·17-s + 18-s + 3.80·19-s − 1.81·20-s + 4.37·21-s − 5.56·22-s − 0.487·23-s + 24-s − 1.71·25-s − 26-s + 27-s + 4.37·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.810·5-s + 0.408·6-s + 1.65·7-s + 0.353·8-s + 0.333·9-s − 0.572·10-s − 1.67·11-s + 0.288·12-s − 0.277·13-s + 1.17·14-s − 0.467·15-s + 0.250·16-s − 1.25·17-s + 0.235·18-s + 0.872·19-s − 0.405·20-s + 0.955·21-s − 1.18·22-s − 0.101·23-s + 0.204·24-s − 0.343·25-s − 0.196·26-s + 0.192·27-s + 0.827·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 1.81T + 5T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 17 | \( 1 + 5.16T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 + 0.487T + 23T^{2} \) |
| 29 | \( 1 + 9.38T + 29T^{2} \) |
| 31 | \( 1 + 0.944T + 31T^{2} \) |
| 37 | \( 1 + 5.87T + 37T^{2} \) |
| 41 | \( 1 - 0.941T + 41T^{2} \) |
| 43 | \( 1 + 8.57T + 43T^{2} \) |
| 47 | \( 1 - 3.30T + 47T^{2} \) |
| 53 | \( 1 + 0.598T + 53T^{2} \) |
| 59 | \( 1 + 8.16T + 59T^{2} \) |
| 61 | \( 1 + 9.67T + 61T^{2} \) |
| 67 | \( 1 - 4.48T + 67T^{2} \) |
| 71 | \( 1 + 3.22T + 71T^{2} \) |
| 73 | \( 1 - 1.48T + 73T^{2} \) |
| 79 | \( 1 - 0.265T + 79T^{2} \) |
| 83 | \( 1 + 5.02T + 83T^{2} \) |
| 89 | \( 1 - 4.65T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−7.52049513990991343256330769509, −7.14966077361388871939095351107, −5.86089894573735830711214369241, −5.10394972002427210058625101272, −4.74603771358647540238705020723, −3.96933793069510068992244912021, −3.17940801305494789916247786381, −2.26115884195020724395712487097, −1.68022883419080830513299684911, 0,
1.68022883419080830513299684911, 2.26115884195020724395712487097, 3.17940801305494789916247786381, 3.96933793069510068992244912021, 4.74603771358647540238705020723, 5.10394972002427210058625101272, 5.86089894573735830711214369241, 7.14966077361388871939095351107, 7.52049513990991343256330769509
