L(s) = 1 | − 2-s − 1.52·3-s + 4-s + 3.51·5-s + 1.52·6-s − 1.62·7-s − 8-s − 0.661·9-s − 3.51·10-s − 5.23·11-s − 1.52·12-s − 0.449·13-s + 1.62·14-s − 5.37·15-s + 16-s + 1.78·17-s + 0.661·18-s − 1.93·19-s + 3.51·20-s + 2.47·21-s + 5.23·22-s − 0.570·23-s + 1.52·24-s + 7.36·25-s + 0.449·26-s + 5.59·27-s − 1.62·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.882·3-s + 0.5·4-s + 1.57·5-s + 0.624·6-s − 0.612·7-s − 0.353·8-s − 0.220·9-s − 1.11·10-s − 1.57·11-s − 0.441·12-s − 0.124·13-s + 0.433·14-s − 1.38·15-s + 0.250·16-s + 0.433·17-s + 0.155·18-s − 0.444·19-s + 0.786·20-s + 0.541·21-s + 1.11·22-s − 0.119·23-s + 0.312·24-s + 1.47·25-s + 0.0881·26-s + 1.07·27-s − 0.306·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 + 1.52T + 3T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 7 | \( 1 + 1.62T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 - 1.78T + 17T^{2} \) |
| 19 | \( 1 + 1.93T + 19T^{2} \) |
| 23 | \( 1 + 0.570T + 23T^{2} \) |
| 29 | \( 1 - 2.33T + 29T^{2} \) |
| 31 | \( 1 - 7.15T + 31T^{2} \) |
| 37 | \( 1 - 0.308T + 37T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 - 8.05T + 43T^{2} \) |
| 47 | \( 1 + 1.84T + 47T^{2} \) |
| 53 | \( 1 + 1.51T + 53T^{2} \) |
| 59 | \( 1 - 0.0896T + 59T^{2} \) |
| 61 | \( 1 + 8.72T + 61T^{2} \) |
| 67 | \( 1 + 2.67T + 67T^{2} \) |
| 71 | \( 1 + 6.78T + 71T^{2} \) |
| 73 | \( 1 + 8.78T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 + 4.78T + 83T^{2} \) |
| 89 | \( 1 - 2.06T + 89T^{2} \) |
| 97 | \( 1 - 0.506T + 97T^{2} \) |
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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
−8.118472245111541666372372198769, −7.32672078058449420385204600173, −6.31193875797395342423555677442, −6.00987601061867723168884734217, −5.41527662612989068552403810800, −4.58048170708254752862762891509, −2.81646307459244043940270841698, −2.55585481000470181138821045455, −1.20900834029276177869573199913, 0,
1.20900834029276177869573199913, 2.55585481000470181138821045455, 2.81646307459244043940270841698, 4.58048170708254752862762891509, 5.41527662612989068552403810800, 6.00987601061867723168884734217, 6.31193875797395342423555677442, 7.32672078058449420385204600173, 8.118472245111541666372372198769