L(s) = 1 | − 2-s − 3-s + 4-s − 3.89·5-s + 6-s + 3.22·7-s − 8-s + 9-s + 3.89·10-s + 11-s − 12-s + 2.07·13-s − 3.22·14-s + 3.89·15-s + 16-s − 1.51·17-s − 18-s − 6.85·19-s − 3.89·20-s − 3.22·21-s − 22-s − 3.58·23-s + 24-s + 10.1·25-s − 2.07·26-s − 27-s + 3.22·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.74·5-s + 0.408·6-s + 1.21·7-s − 0.353·8-s + 0.333·9-s + 1.23·10-s + 0.301·11-s − 0.288·12-s + 0.575·13-s − 0.861·14-s + 1.00·15-s + 0.250·16-s − 0.366·17-s − 0.235·18-s − 1.57·19-s − 0.871·20-s − 0.703·21-s − 0.213·22-s − 0.748·23-s + 0.204·24-s + 2.03·25-s − 0.407·26-s − 0.192·27-s + 0.608·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 + 3.89T + 5T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 13 | \( 1 - 2.07T + 13T^{2} \) |
| 17 | \( 1 + 1.51T + 17T^{2} \) |
| 19 | \( 1 + 6.85T + 19T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 - 5.44T + 29T^{2} \) |
| 31 | \( 1 + 2.54T + 31T^{2} \) |
| 37 | \( 1 - 1.72T + 37T^{2} \) |
| 41 | \( 1 - 1.77T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 + 8.69T + 47T^{2} \) |
| 53 | \( 1 - 0.850T + 53T^{2} \) |
| 59 | \( 1 - 0.453T + 59T^{2} \) |
| 67 | \( 1 + 3.84T + 67T^{2} \) |
| 71 | \( 1 - 4.50T + 71T^{2} \) |
| 73 | \( 1 + 3.75T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 4.65T + 83T^{2} \) |
| 89 | \( 1 - 2.59T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.261441745507353623885103095291, −7.53901249219183960151473632762, −6.75190774695576553434756634523, −6.08390544133764622759027839480, −4.85673290011215092494825493424, −4.31067021768206019319715827589, −3.60558767344153025746327592912, −2.22209271726599498837237613990, −1.10726875054537707796267785005, 0,
1.10726875054537707796267785005, 2.22209271726599498837237613990, 3.60558767344153025746327592912, 4.31067021768206019319715827589, 4.85673290011215092494825493424, 6.08390544133764622759027839480, 6.75190774695576553434756634523, 7.53901249219183960151473632762, 8.261441745507353623885103095291