L(s) = 1 | + 2-s − 0.585·3-s + 4-s + 5-s − 0.585·6-s − 4.41·7-s + 8-s − 2.65·9-s + 10-s − 5·11-s − 0.585·12-s − 6.24·13-s − 4.41·14-s − 0.585·15-s + 16-s + 4.24·17-s − 2.65·18-s − 4.82·19-s + 20-s + 2.58·21-s − 5·22-s + 5.65·23-s − 0.585·24-s + 25-s − 6.24·26-s + 3.31·27-s − 4.41·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.338·3-s + 0.5·4-s + 0.447·5-s − 0.239·6-s − 1.66·7-s + 0.353·8-s − 0.885·9-s + 0.316·10-s − 1.50·11-s − 0.169·12-s − 1.73·13-s − 1.17·14-s − 0.151·15-s + 0.250·16-s + 1.02·17-s − 0.626·18-s − 1.10·19-s + 0.223·20-s + 0.564·21-s − 1.06·22-s + 1.17·23-s − 0.119·24-s + 0.200·25-s − 1.22·26-s + 0.637·27-s − 0.834·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 67 | \( 1 + T \) |
good | 3 | \( 1 + 0.585T + 3T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 - 2.58T + 31T^{2} \) |
| 37 | \( 1 + 8.41T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 3.75T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 9.41T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 - 6.41T + 61T^{2} \) |
| 71 | \( 1 - 0.757T + 71T^{2} \) |
| 73 | \( 1 + 5.07T + 73T^{2} \) |
| 79 | \( 1 + 3.65T + 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 + 4.17T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−10.19947734742649570046079944655, −9.496434924194965227078495229984, −8.243389625678056073782624889020, −7.12929380060670577790837201830, −6.35005596013460418721474379894, −5.48545555777389928545223906482, −4.78779295211955270809225358042, −3.01275932430421438995362219170, −2.68964529335699586784626203640, 0,
2.68964529335699586784626203640, 3.01275932430421438995362219170, 4.78779295211955270809225358042, 5.48545555777389928545223906482, 6.35005596013460418721474379894, 7.12929380060670577790837201830, 8.243389625678056073782624889020, 9.496434924194965227078495229984, 10.19947734742649570046079944655