L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.334 + 0.334i)5-s − 4.55i·7-s − 1.00i·9-s + (−2.47 − 2.47i)11-s + (0.0594 − 0.0594i)13-s − 0.473·15-s + 3.61·17-s + (2.55 − 2.55i)19-s + (3.22 + 3.22i)21-s + 2.82i·23-s − 4.77i·25-s + (0.707 + 0.707i)27-s + (5.16 − 5.16i)29-s + 0.557·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.149 + 0.149i)5-s − 1.72i·7-s − 0.333i·9-s + (−0.745 − 0.745i)11-s + (0.0164 − 0.0164i)13-s − 0.122·15-s + 0.877·17-s + (0.586 − 0.586i)19-s + (0.703 + 0.703i)21-s + 0.589i·23-s − 0.955i·25-s + (0.136 + 0.136i)27-s + (0.958 − 0.958i)29-s + 0.100·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.939952 - 0.550363i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.939952 - 0.550363i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
good | 5 | \( 1 + (-0.334 - 0.334i)T + 5iT^{2} \) |
| 7 | \( 1 + 4.55iT - 7T^{2} \) |
| 11 | \( 1 + (2.47 + 2.47i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.0594 + 0.0594i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 + (-2.55 + 2.55i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (-5.16 + 5.16i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.557T + 31T^{2} \) |
| 37 | \( 1 + (4.38 + 4.38i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.27iT - 41T^{2} \) |
| 43 | \( 1 + (1.61 + 1.61i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (-0.493 - 0.493i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4 - 4i)T + 59iT^{2} \) |
| 61 | \( 1 + (2.72 - 2.72i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3.77 + 3.77i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.11iT - 71T^{2} \) |
| 73 | \( 1 - 0.541iT - 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + (10.6 - 10.6i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−11.01695633079701709036634198539, −10.28693071552471284761898725833, −9.751154433522731144514845492981, −8.263825234074577498598324203190, −7.42927515068589563340550320691, −6.41329535997847329054254256878, −5.27062066572200582375536195055, −4.19294989693500986486603577566, −3.09915399267279008214102448632, −0.798323976232333238655454651688,
1.81941199671781103511855085726, 3.06489859632359584099818803741, 5.06580784456107007101824914050, 5.52705341607549354773506541956, 6.67163230667771587395432143200, 7.81903369939202606960027400202, 8.690843359914243989703677178462, 9.668021172584295950282863469873, 10.57477736703521386096885961694, 11.80772335294602021402030410493