| L(s) = 1 | − 1.54·2-s − 3-s + 0.392·4-s + 3.59·5-s + 1.54·6-s − 3.93·7-s + 2.48·8-s + 9-s − 5.55·10-s + 3.85·11-s − 0.392·12-s − 1.41·13-s + 6.08·14-s − 3.59·15-s − 4.63·16-s − 17-s − 1.54·18-s − 0.145·19-s + 1.40·20-s + 3.93·21-s − 5.96·22-s − 0.400·23-s − 2.48·24-s + 7.89·25-s + 2.19·26-s − 27-s − 1.54·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s − 0.577·3-s + 0.196·4-s + 1.60·5-s + 0.631·6-s − 1.48·7-s + 0.879·8-s + 0.333·9-s − 1.75·10-s + 1.16·11-s − 0.113·12-s − 0.393·13-s + 1.62·14-s − 0.927·15-s − 1.15·16-s − 0.242·17-s − 0.364·18-s − 0.0334·19-s + 0.314·20-s + 0.859·21-s − 1.27·22-s − 0.0835·23-s − 0.507·24-s + 1.57·25-s + 0.430·26-s − 0.192·27-s − 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.54T + 2T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 11 | \( 1 - 3.85T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 19 | \( 1 + 0.145T + 19T^{2} \) |
| 23 | \( 1 + 0.400T + 23T^{2} \) |
| 29 | \( 1 + 4.77T + 29T^{2} \) |
| 31 | \( 1 - 9.59T + 31T^{2} \) |
| 37 | \( 1 + 8.11T + 37T^{2} \) |
| 41 | \( 1 - 2.99T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 + 8.05T + 47T^{2} \) |
| 53 | \( 1 - 4.69T + 53T^{2} \) |
| 59 | \( 1 + 8.98T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 5.49T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 6.38T + 73T^{2} \) |
| 79 | \( 1 - 8.76T + 79T^{2} \) |
| 83 | \( 1 + 6.68T + 83T^{2} \) |
| 89 | \( 1 + 7.17T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.37111061201146515709336399796, −6.57287184098402268886742729629, −6.43909187387008055911434753816, −5.62435933649499824284154960250, −4.80329228291280260513482407726, −3.91672648730122424102293378486, −2.85336300688698370698262449294, −1.88762568286239313033461501873, −1.12664855102117093871172896802, 0,
1.12664855102117093871172896802, 1.88762568286239313033461501873, 2.85336300688698370698262449294, 3.91672648730122424102293378486, 4.80329228291280260513482407726, 5.62435933649499824284154960250, 6.43909187387008055911434753816, 6.57287184098402268886742729629, 7.37111061201146515709336399796
