L(s) = 1 | − 2-s − 1.74·3-s + 4-s + 5-s + 1.74·6-s + 2.79·7-s − 8-s + 0.0402·9-s − 10-s + 2.90·11-s − 1.74·12-s + 3.29·13-s − 2.79·14-s − 1.74·15-s + 16-s − 4.26·17-s − 0.0402·18-s − 0.133·19-s + 20-s − 4.88·21-s − 2.90·22-s − 1.71·23-s + 1.74·24-s + 25-s − 3.29·26-s + 5.16·27-s + 2.79·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.00·3-s + 0.5·4-s + 0.447·5-s + 0.711·6-s + 1.05·7-s − 0.353·8-s + 0.0134·9-s − 0.316·10-s + 0.876·11-s − 0.503·12-s + 0.914·13-s − 0.748·14-s − 0.450·15-s + 0.250·16-s − 1.03·17-s − 0.00948·18-s − 0.0305·19-s + 0.223·20-s − 1.06·21-s − 0.619·22-s − 0.356·23-s + 0.355·24-s + 0.200·25-s − 0.646·26-s + 0.993·27-s + 0.528·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 1.74T + 3T^{2} \) |
| 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 13 | \( 1 - 3.29T + 13T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 + 0.133T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 + 2.64T + 29T^{2} \) |
| 31 | \( 1 + 9.13T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 + 5.26T + 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 - 6.49T + 47T^{2} \) |
| 53 | \( 1 - 4.03T + 53T^{2} \) |
| 59 | \( 1 + 6.96T + 59T^{2} \) |
| 61 | \( 1 + 5.75T + 61T^{2} \) |
| 67 | \( 1 + 1.15T + 67T^{2} \) |
| 71 | \( 1 + 0.443T + 71T^{2} \) |
| 73 | \( 1 + 0.955T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 8.57T + 97T^{2} \) |
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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
−8.310931264852036419769249865584, −7.19544657993500368293944926705, −6.65965964136783610259906346935, −5.87103839757841456013532536133, −5.33146196045535496981857754241, −4.41206555336677188955922393837, −3.41823327839763969706561801379, −1.95860501700622797059490697609, −1.39853409528883700262324910648, 0,
1.39853409528883700262324910648, 1.95860501700622797059490697609, 3.41823327839763969706561801379, 4.41206555336677188955922393837, 5.33146196045535496981857754241, 5.87103839757841456013532536133, 6.65965964136783610259906346935, 7.19544657993500368293944926705, 8.310931264852036419769249865584