| L(s) = 1 | + (1.73 + 1.73i)2-s + 3.99i·4-s + (1.22 − 1.22i)7-s + (−3.46 + 3.46i)8-s + 4.24i·11-s + (−3.67 − 3.67i)13-s + 4.24·14-s − 3.99·16-s + (−1.73 − 1.73i)17-s − 5i·19-s + (−7.34 + 7.34i)22-s + (1.73 − 1.73i)23-s − 12.7i·26-s + (4.89 + 4.89i)28-s + 4.24·29-s + ⋯ |
| L(s) = 1 | + (1.22 + 1.22i)2-s + 1.99i·4-s + (0.462 − 0.462i)7-s + (−1.22 + 1.22i)8-s + 1.27i·11-s + (−1.01 − 1.01i)13-s + 1.13·14-s − 0.999·16-s + (−0.420 − 0.420i)17-s − 1.14i·19-s + (−1.56 + 1.56i)22-s + (0.361 − 0.361i)23-s − 2.49i·26-s + (0.925 + 0.925i)28-s + 0.787·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42254 + 1.67489i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42254 + 1.67489i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-1.73 - 1.73i)T + 2iT^{2} \) |
| 7 | \( 1 + (-1.22 + 1.22i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 13 | \( 1 + (3.67 + 3.67i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.73 + 1.73i)T + 17iT^{2} \) |
| 19 | \( 1 + 5iT - 19T^{2} \) |
| 23 | \( 1 + (-1.73 + 1.73i)T - 23iT^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 + (1.22 + 1.22i)T + 43iT^{2} \) |
| 47 | \( 1 + (5.19 + 5.19i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.92 - 6.92i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + (-3.67 + 3.67i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 + (-2.44 - 2.44i)T + 73iT^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 + (-1.73 + 1.73i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.48T + 89T^{2} \) |
| 97 | \( 1 + (8.57 - 8.57i)T - 97iT^{2} \) |
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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
−12.76326097835072108155974907559, −12.03836381489548060961156198571, −10.70833690122989171497862312399, −9.459379223035889961518583690411, −7.963455720396301469773036376915, −7.31977083729115718513766763869, −6.43632609983115554070979723455, −4.88639721950486722783021450170, −4.63472054137443077423453915682, −2.85391935217092437432548227444,
1.79318744664388340199161078881, 3.11484971180110186144680182269, 4.34702313679672812887203264441, 5.36868942085162070229423723846, 6.40061017556961704003843862937, 8.145481342643812868966019063657, 9.368068875813559934955589780812, 10.45506297873959919173887657904, 11.35006249867237928513604936421, 11.94759769507453077428932928475
