| L(s) = 1 | + i·2-s − i·3-s − 4-s + 3.45i·5-s + 6-s − i·8-s − 9-s − 3.45·10-s + (3.07 − 1.24i)11-s + i·12-s + 5.39·13-s + 3.45·15-s + 16-s − 0.0457·17-s − i·18-s + 6.95·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 1.54i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s − 1.09·10-s + (0.927 − 0.373i)11-s + 0.288i·12-s + 1.49·13-s + 0.892·15-s + 0.250·16-s − 0.0110·17-s − 0.235i·18-s + 1.59·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.091581507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091581507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.07 + 1.24i)T \) |
| good | 5 | \( 1 - 3.45iT - 5T^{2} \) |
| 13 | \( 1 - 5.39T + 13T^{2} \) |
| 17 | \( 1 + 0.0457T + 17T^{2} \) |
| 19 | \( 1 - 6.95T + 19T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 + 7.78iT - 29T^{2} \) |
| 31 | \( 1 + 6.61iT - 31T^{2} \) |
| 37 | \( 1 + 1.51T + 37T^{2} \) |
| 41 | \( 1 + 4.87T + 41T^{2} \) |
| 43 | \( 1 - 3.27iT - 43T^{2} \) |
| 47 | \( 1 + 12.1iT - 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 + 14.8iT - 59T^{2} \) |
| 61 | \( 1 - 3.63T + 61T^{2} \) |
| 67 | \( 1 - 6.66T + 67T^{2} \) |
| 71 | \( 1 - 6.89T + 71T^{2} \) |
| 73 | \( 1 - 3.04T + 73T^{2} \) |
| 79 | \( 1 - 12.9iT - 79T^{2} \) |
| 83 | \( 1 - 8.48T + 83T^{2} \) |
| 89 | \( 1 + 4.20iT - 89T^{2} \) |
| 97 | \( 1 - 6.64iT - 97T^{2} \) |
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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
−8.478046330725175971838559143623, −7.87600550168316960830734014456, −7.13708265334992295724656093880, −6.43680908217798936480074513511, −6.16718346251924452599460157567, −5.22954088009627021757645875665, −3.71158413964076275659709442935, −3.46747921055343505616309462568, −2.19885812467814505294296516991, −0.870895432307550616293089466724,
1.06134369362510512676075005300, 1.50683459099591271580719923430, 3.20789645421059400890695508753, 3.78892227550673802340076075754, 4.69940597181516334897813789640, 5.19645169838305976824097228570, 6.02374855872989588369493753606, 7.13775545263898630991053060737, 8.230067885286761341996259608872, 8.894013786068450342813728375665
