L(s) = 1 | + (−1.66 − 1.66i)2-s + 3.56i·4-s + (0.420 + 2.19i)5-s + (0.707 − 0.707i)7-s + (2.60 − 2.60i)8-s + (2.96 − 4.36i)10-s + 3.90i·11-s + (3.99 + 3.99i)13-s − 2.35·14-s − 1.57·16-s + (−5.52 − 5.52i)17-s + 2.04i·19-s + (−7.82 + 1.49i)20-s + (6.51 − 6.51i)22-s + (−0.704 + 0.704i)23-s + ⋯ |
L(s) = 1 | + (−1.17 − 1.17i)2-s + 1.78i·4-s + (0.188 + 0.982i)5-s + (0.267 − 0.267i)7-s + (0.922 − 0.922i)8-s + (0.936 − 1.38i)10-s + 1.17i·11-s + (1.10 + 1.10i)13-s − 0.630·14-s − 0.394·16-s + (−1.34 − 1.34i)17-s + 0.469i·19-s + (−1.75 + 0.335i)20-s + (1.38 − 1.38i)22-s + (−0.146 + 0.146i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.464374 + 0.319910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.464374 + 0.319910i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.420 - 2.19i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (1.66 + 1.66i)T + 2iT^{2} \) |
| 11 | \( 1 - 3.90iT - 11T^{2} \) |
| 13 | \( 1 + (-3.99 - 3.99i)T + 13iT^{2} \) |
| 17 | \( 1 + (5.52 + 5.52i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.04iT - 19T^{2} \) |
| 23 | \( 1 + (0.704 - 0.704i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 + (-2.61 + 2.61i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.05iT - 41T^{2} \) |
| 43 | \( 1 + (4.64 + 4.64i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.17 - 5.17i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.29 - 4.29i)T - 53iT^{2} \) |
| 59 | \( 1 + 6.73T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + (2.23 - 2.23i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.03iT - 71T^{2} \) |
| 73 | \( 1 + (-8.12 - 8.12i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.60iT - 79T^{2} \) |
| 83 | \( 1 + (-9.62 + 9.62i)T - 83iT^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + (1.81 - 1.81i)T - 97iT^{2} \) |
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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
−10.26369666449022936924230932053, −9.348208831710606268607647974180, −9.058353971593316202005571846392, −7.73680950458498819764047250295, −7.16200269333844370119973665182, −6.20107849561355435311689696461, −4.53911397445357665066743361598, −3.56303901182979509171425691042, −2.37130623432651757138260214837, −1.63162397889644792828473986647,
0.39527718440861657168424295576, 1.66788738494377180221472923606, 3.62901801616905630392651842049, 5.03448096456727818001422817641, 5.95573696915417639160079233627, 6.30086286018076400047683822675, 7.71158819070617789348730208707, 8.378109938199571265034823962658, 8.765385483321468845555295322809, 9.407159705093299985161845646067