L(s) = 1 | + (−0.707 + 0.707i)2-s + (−1.68 − 0.398i)3-s − 1.00i·4-s + (2.32 − 2.32i)5-s + (1.47 − 0.910i)6-s + (−2.38 + 2.38i)7-s + (0.707 + 0.707i)8-s + (2.68 + 1.34i)9-s + 3.29i·10-s + (2.36 + 2.36i)11-s + (−0.398 + 1.68i)12-s − 3.37i·14-s + (−4.85 + 2.99i)15-s − 1.00·16-s − 0.142·17-s + (−2.84 + 0.948i)18-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.973 − 0.229i)3-s − 0.500i·4-s + (1.04 − 1.04i)5-s + (0.601 − 0.371i)6-s + (−0.901 + 0.901i)7-s + (0.250 + 0.250i)8-s + (0.894 + 0.447i)9-s + 1.04i·10-s + (0.714 + 0.714i)11-s + (−0.114 + 0.486i)12-s − 0.901i·14-s + (−1.25 + 0.774i)15-s − 0.250·16-s − 0.0344·17-s + (−0.670 + 0.223i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8572077832\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8572077832\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (1.68 + 0.398i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + (-2.32 + 2.32i)T - 5iT^{2} \) |
| 7 | \( 1 + (2.38 - 2.38i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.36 - 2.36i)T + 11iT^{2} \) |
| 17 | \( 1 + 0.142T + 17T^{2} \) |
| 19 | \( 1 + (0.900 + 0.900i)T + 19iT^{2} \) |
| 23 | \( 1 + 7.27T + 23T^{2} \) |
| 29 | \( 1 - 8.53iT - 29T^{2} \) |
| 31 | \( 1 + (-4.04 - 4.04i)T + 31iT^{2} \) |
| 37 | \( 1 + (-8.22 + 8.22i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.570 - 0.570i)T - 41iT^{2} \) |
| 43 | \( 1 - 4.19iT - 43T^{2} \) |
| 47 | \( 1 + (-3.90 - 3.90i)T + 47iT^{2} \) |
| 53 | \( 1 - 8.81iT - 53T^{2} \) |
| 59 | \( 1 + (-7.10 - 7.10i)T + 59iT^{2} \) |
| 61 | \( 1 - 0.696T + 61T^{2} \) |
| 67 | \( 1 + (-0.435 - 0.435i)T + 67iT^{2} \) |
| 71 | \( 1 + (-5.76 + 5.76i)T - 71iT^{2} \) |
| 73 | \( 1 + (1.54 - 1.54i)T - 73iT^{2} \) |
| 79 | \( 1 + 4.42T + 79T^{2} \) |
| 83 | \( 1 + (1.61 - 1.61i)T - 83iT^{2} \) |
| 89 | \( 1 + (-5.45 - 5.45i)T + 89iT^{2} \) |
| 97 | \( 1 + (-5.33 - 5.33i)T + 97iT^{2} \) |
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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
−9.843252686209796291182469926115, −9.349150920460927025545005347714, −8.700902461888641343968853608655, −7.47517736889039375227416976556, −6.43900299768123254976784508996, −6.00265945375118985759954459188, −5.25513560787437723430972790284, −4.32032462840983728038254949573, −2.25554713488489692985141641092, −1.17131948609106417570073484835,
0.60088281522456618473409003291, 2.15748985783108543572039096789, 3.47539747416364412519666723743, 4.24256210583224779324013144470, 5.92736798744743152318130534203, 6.32042109968448672013953443406, 7.01415973117968843309767686100, 8.162878739449012152412860065663, 9.562094765215709175252669052630, 10.03413093222183648157423264243