L(s) = 1 | + 1.89·2-s − 0.392·3-s + 1.58·4-s − 4.13·5-s − 0.743·6-s + 2.17·7-s − 0.788·8-s − 2.84·9-s − 7.83·10-s − 2.90·11-s − 0.621·12-s + 3.60·13-s + 4.11·14-s + 1.62·15-s − 4.65·16-s − 6.33·17-s − 5.38·18-s − 2.54·19-s − 6.55·20-s − 0.853·21-s − 5.49·22-s − 23-s + 0.309·24-s + 12.1·25-s + 6.82·26-s + 2.29·27-s + 3.44·28-s + ⋯ |
L(s) = 1 | + 1.33·2-s − 0.226·3-s + 0.791·4-s − 1.84·5-s − 0.303·6-s + 0.821·7-s − 0.278·8-s − 0.948·9-s − 2.47·10-s − 0.875·11-s − 0.179·12-s + 0.999·13-s + 1.09·14-s + 0.419·15-s − 1.16·16-s − 1.53·17-s − 1.26·18-s − 0.582·19-s − 1.46·20-s − 0.186·21-s − 1.17·22-s − 0.208·23-s + 0.0631·24-s + 2.42·25-s + 1.33·26-s + 0.441·27-s + 0.650·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 3 | \( 1 + 0.392T + 3T^{2} \) |
| 5 | \( 1 + 4.13T + 5T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 + 6.33T + 17T^{2} \) |
| 19 | \( 1 + 2.54T + 19T^{2} \) |
| 31 | \( 1 + 0.623T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 + 5.90T + 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 - 9.54T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 6.14T + 59T^{2} \) |
| 61 | \( 1 - 0.927T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 5.20T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 0.490T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 7.45T + 97T^{2} \) |
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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
−10.83640345172142218808346902901, −8.664743869117325586470043910270, −8.456553953644733448478392774280, −7.32764094441775499362495463816, −6.28893042611169511617243990056, −5.22746853888916750330038609747, −4.43505416213874845271410972039, −3.74339274849255335300092625221, −2.61920845897709971750264121001, 0,
2.61920845897709971750264121001, 3.74339274849255335300092625221, 4.43505416213874845271410972039, 5.22746853888916750330038609747, 6.28893042611169511617243990056, 7.32764094441775499362495463816, 8.456553953644733448478392774280, 8.664743869117325586470043910270, 10.83640345172142218808346902901