L(s) = 1 | + 2-s + (−1.67 − 0.423i)3-s + 4-s + i·5-s + (−1.67 − 0.423i)6-s − 2.47·7-s + 8-s + (2.64 + 1.42i)9-s + i·10-s + 2i·11-s + (−1.67 − 0.423i)12-s + 1.15i·13-s − 2.47·14-s + (0.423 − 1.67i)15-s + 16-s + 6.67i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.969 − 0.244i)3-s + 0.5·4-s + 0.447i·5-s + (−0.685 − 0.173i)6-s − 0.934·7-s + 0.353·8-s + (0.880 + 0.474i)9-s + 0.316i·10-s + 0.603i·11-s + (−0.484 − 0.122i)12-s + 0.319i·13-s − 0.660·14-s + (0.109 − 0.433i)15-s + 0.250·16-s + 1.61i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04048 + 0.803032i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04048 + 0.803032i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.67 + 0.423i)T \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 + (-4.35 - 0.0388i)T \) |
good | 7 | \( 1 + 2.47T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 1.15iT - 13T^{2} \) |
| 17 | \( 1 - 6.67iT - 17T^{2} \) |
| 23 | \( 1 - 5.79iT - 23T^{2} \) |
| 29 | \( 1 + 3.35T + 29T^{2} \) |
| 31 | \( 1 + 4.80iT - 31T^{2} \) |
| 37 | \( 1 - 11.6iT - 37T^{2} \) |
| 41 | \( 1 - 7.83T + 41T^{2} \) |
| 43 | \( 1 + 0.978T + 43T^{2} \) |
| 47 | \( 1 + 3.02iT - 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 10.0iT - 67T^{2} \) |
| 71 | \( 1 - 7.24T + 71T^{2} \) |
| 73 | \( 1 + 6.97T + 73T^{2} \) |
| 79 | \( 1 - 5.75iT - 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 6.94iT - 97T^{2} \) |
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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
−11.15228646240616272409950074580, −10.14295071813432284128252387452, −9.583945568990443122885005579190, −7.86799882297431554592093231524, −7.03651509535782925320400516677, −6.24923638935148400568156792013, −5.58094114090127515657353920110, −4.34351932684991148888378980798, −3.32951930052659455684728004466, −1.72909132828000856978625719889,
0.68321368082771928698069066745, 2.86169804252202260323718457170, 3.99615310095181805249574398042, 5.11115050798851559239319262616, 5.74482728870097784554621279933, 6.71382535436016097414201439506, 7.53070806398092478785569219503, 9.068760370269626693718312067199, 9.769509726449168326464905158652, 10.77950656023800204584367987937