L(s) = 1 | + (1.60 + 0.649i)3-s + (−0.468 − 0.811i)5-s + (0.5 − 0.866i)7-s + (2.15 + 2.08i)9-s + (2.48 − 4.30i)11-s + (−0.622 − 1.07i)13-s + (−0.225 − 1.60i)15-s + 5.22·17-s − 5.18·19-s + (1.36 − 1.06i)21-s + (−1.00 − 1.73i)23-s + (2.06 − 3.57i)25-s + (2.10 + 4.74i)27-s + (−3.43 + 5.95i)29-s + (−2.86 − 4.96i)31-s + ⋯ |
L(s) = 1 | + (0.927 + 0.374i)3-s + (−0.209 − 0.362i)5-s + (0.188 − 0.327i)7-s + (0.718 + 0.695i)9-s + (0.749 − 1.29i)11-s + (−0.172 − 0.298i)13-s + (−0.0581 − 0.414i)15-s + 1.26·17-s − 1.18·19-s + (0.297 − 0.232i)21-s + (−0.209 − 0.362i)23-s + (0.412 − 0.714i)25-s + (0.405 + 0.913i)27-s + (−0.638 + 1.10i)29-s + (−0.514 − 0.891i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.277735950\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.277735950\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.60 - 0.649i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.468 + 0.811i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.48 + 4.30i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.622 + 1.07i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 + (1.00 + 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.43 - 5.95i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.86 + 4.96i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.73T + 37T^{2} \) |
| 41 | \( 1 + (-5.73 - 9.93i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.80 + 8.31i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.984 - 1.70i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 7.63T + 53T^{2} \) |
| 59 | \( 1 + (-2.43 - 4.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.52 + 2.63i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.573 + 0.994i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.83T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 + (6.05 - 10.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.431 - 0.747i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + (3.78 - 6.55i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.746954332143724002072719170298, −9.041810580685306115199939893101, −8.217121071004584759584339007414, −7.78184134377494251563916804782, −6.54876907001369758443615649599, −5.53119220913855343133653045655, −4.35090966548687044127653355724, −3.68596881766531208926260529681, −2.61862361815010886389404078914, −1.07576780015636935545019452205,
1.55429264845187736571633896404, 2.51417421035978497324770397266, 3.71681788162767728136194258438, 4.49166626005601476520030052765, 5.88860527578448593865401029384, 6.89791231897976564404374116807, 7.52362871823919186202933656638, 8.253253151281955180733714447938, 9.399968942385806483774155750637, 9.613313139476417931860186211483