L(s) = 1 | + (0.951 − 0.309i)3-s + 1.04i·7-s + (0.809 − 0.587i)9-s + (−5.08 − 3.69i)11-s + (−0.591 − 0.814i)13-s + (−4.46 − 1.44i)17-s + (1.84 − 5.68i)19-s + (0.323 + 0.995i)21-s + (−4.73 + 6.51i)23-s + (0.587 − 0.809i)27-s + (−2.13 − 6.57i)29-s + (−2.94 + 9.05i)31-s + (−5.98 − 1.94i)33-s + (−4.52 − 6.22i)37-s + (−0.814 − 0.591i)39-s + ⋯ |
L(s) = 1 | + (0.549 − 0.178i)3-s + 0.395i·7-s + (0.269 − 0.195i)9-s + (−1.53 − 1.11i)11-s + (−0.164 − 0.225i)13-s + (−1.08 − 0.351i)17-s + (0.423 − 1.30i)19-s + (0.0705 + 0.217i)21-s + (−0.986 + 1.35i)23-s + (0.113 − 0.155i)27-s + (−0.396 − 1.22i)29-s + (−0.528 + 1.62i)31-s + (−1.04 − 0.338i)33-s + (−0.743 − 1.02i)37-s + (−0.130 − 0.0947i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8454732665\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8454732665\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.04iT - 7T^{2} \) |
| 11 | \( 1 + (5.08 + 3.69i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.591 + 0.814i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (4.46 + 1.44i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.84 + 5.68i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (4.73 - 6.51i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.13 + 6.57i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.94 - 9.05i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.52 + 6.22i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.26 - 0.922i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 9.94iT - 43T^{2} \) |
| 47 | \( 1 + (-4.60 + 1.49i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.68 + 0.873i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.30 + 2.39i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (2.55 + 1.85i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.01 - 0.979i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.01 - 6.19i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.216 + 0.297i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.02 - 3.15i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (13.1 + 4.28i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (2.02 + 1.47i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.173 - 0.0564i)T + (78.4 - 57.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.952160809048354137255312717002, −8.507952998277873616149391186368, −7.57065826018876271931589096589, −6.96528581741525014319418134351, −5.66039185717572045583202519442, −5.22597317166061893427473363486, −3.87819198612147514873686138384, −2.88336107028274259369813216193, −2.14165901893921645328204072377, −0.28058888739357635943390640975,
1.85320036099457060881917933373, 2.66488655265890231690283245319, 3.96098775970669530129488653612, 4.61328567677544448008270202607, 5.60911438642637387894447854682, 6.70032979076385453792475022598, 7.58233478479341983421771386675, 8.077213730773683627198050185066, 8.980751460281357576852993174820, 9.984950522083517302501911109549