L(s) = 1 | + 2.26·2-s − 3-s + 3.11·4-s − 2.97·5-s − 2.26·6-s + 2.11·7-s + 2.51·8-s + 9-s − 6.73·10-s − 1.03·11-s − 3.11·12-s − 2.11·13-s + 4.78·14-s + 2.97·15-s − 0.543·16-s + 17-s + 2.26·18-s + 3.83·19-s − 9.26·20-s − 2.11·21-s − 2.35·22-s − 0.935·23-s − 2.51·24-s + 3.86·25-s − 4.78·26-s − 27-s + 6.58·28-s + ⋯ |
L(s) = 1 | + 1.59·2-s − 0.577·3-s + 1.55·4-s − 1.33·5-s − 0.922·6-s + 0.799·7-s + 0.887·8-s + 0.333·9-s − 2.12·10-s − 0.313·11-s − 0.898·12-s − 0.586·13-s + 1.27·14-s + 0.768·15-s − 0.135·16-s + 0.242·17-s + 0.532·18-s + 0.879·19-s − 2.07·20-s − 0.461·21-s − 0.501·22-s − 0.194·23-s − 0.512·24-s + 0.773·25-s − 0.937·26-s − 0.192·27-s + 1.24·28-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 | \( 1 - 2.26T + 2T^{2} \) |
| 5 | \( 1 + 2.97T + 5T^{2} \) |
| 7 | \( 1 - 2.11T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 2.11T + 13T^{2} \) |
| 19 | \( 1 - 3.83T + 19T^{2} \) |
| 23 | \( 1 + 0.935T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 + 0.803T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 + 0.289T + 43T^{2} \) |
| 47 | \( 1 + 0.822T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 + 2.76T + 67T^{2} \) |
| 71 | \( 1 - 0.913T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 9.46T + 89T^{2} \) |
| 97 | \( 1 + 5.55T + 97T^{2} \) |
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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
−7.27988618457011812424830605300, −6.76160770971137663732430033527, −5.78452075624230493203173157278, −5.23419070830918789442609915537, −4.59847309709118840364433704548, −4.18962252077684848726401229887, −3.32767007397429022253572456367, −2.64472579317572574284736273410, −1.39667391426339235262191607308, 0,
1.39667391426339235262191607308, 2.64472579317572574284736273410, 3.32767007397429022253572456367, 4.18962252077684848726401229887, 4.59847309709118840364433704548, 5.23419070830918789442609915537, 5.78452075624230493203173157278, 6.76160770971137663732430033527, 7.27988618457011812424830605300