L(s) = 1 | − 1.73·3-s + 2i·4-s + (2.09 − 2.09i)7-s + 2.99·9-s − 3.46i·12-s − 4·16-s + (5.73 + 5.73i)19-s + (−3.63 + 3.63i)21-s + 5i·25-s − 5.19·27-s + (4.19 + 4.19i)28-s + (7.83 + 7.83i)31-s + 5.99i·36-s + (1.53 − 1.53i)37-s + 1.73i·43-s + ⋯ |
L(s) = 1 | − 1.00·3-s + i·4-s + (0.792 − 0.792i)7-s + 0.999·9-s − 0.999i·12-s − 16-s + (1.31 + 1.31i)19-s + (−0.792 + 0.792i)21-s + i·25-s − 1.00·27-s + (0.792 + 0.792i)28-s + (1.40 + 1.40i)31-s + 0.999i·36-s + (0.252 − 0.252i)37-s + 0.264i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.981709 + 0.533611i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.981709 + 0.533611i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2iT^{2} \) |
| 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 + (-2.09 + 2.09i)T - 7iT^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (-5.73 - 5.73i)T + 19iT^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (-7.83 - 7.83i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.53 + 1.53i)T - 37iT^{2} \) |
| 41 | \( 1 - 41iT^{2} \) |
| 43 | \( 1 - 1.73iT - 43T^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + 8.66T + 61T^{2} \) |
| 67 | \( 1 + (0.562 + 0.562i)T + 67iT^{2} \) |
| 71 | \( 1 - 71iT^{2} \) |
| 73 | \( 1 + (-9.36 + 9.36i)T - 73iT^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 89iT^{2} \) |
| 97 | \( 1 + (12.0 + 12.0i)T + 97iT^{2} \) |
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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
−11.14913337958673341010578189675, −10.39421141873949194170659792561, −9.370181682279058816630658204357, −8.024909707748857791448514343863, −7.51258809412948620450992386380, −6.57659810474222086040522359080, −5.29947676156940474400782447834, −4.40671778278207042767954334882, −3.37122348620860541365737243198, −1.37712763592021598865447856777,
0.894111108031332083036666042619, 2.36098811863784298264928976605, 4.50450951165875432678108561816, 5.17541792611726526999647179595, 5.97955408574736551149295518241, 6.84813306304365579723562258429, 8.036085470795560841555742943543, 9.264710070122386125204307351442, 9.947119955993524784298996576903, 10.92683057406370130174396056950