L(s) = 1 | + (−0.765 − 0.765i)3-s + (−0.414 + 0.414i)5-s + 3.69i·7-s − 1.82i·9-s + (−2.93 + 2.93i)11-s + (−2.41 − 2.41i)13-s + 0.634·15-s − 2.82·17-s + (−4.46 − 4.46i)19-s + (2.82 − 2.82i)21-s + 6.75i·23-s + 4.65i·25-s + (−3.69 + 3.69i)27-s + (5.24 + 5.24i)29-s − 3.06·31-s + ⋯ |
L(s) = 1 | + (−0.441 − 0.441i)3-s + (−0.185 + 0.185i)5-s + 1.39i·7-s − 0.609i·9-s + (−0.883 + 0.883i)11-s + (−0.669 − 0.669i)13-s + 0.163·15-s − 0.685·17-s + (−1.02 − 1.02i)19-s + (0.617 − 0.617i)21-s + 1.40i·23-s + 0.931i·25-s + (−0.711 + 0.711i)27-s + (0.973 + 0.973i)29-s − 0.549·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.160308 + 0.387019i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.160308 + 0.387019i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.765 + 0.765i)T + 3iT^{2} \) |
| 5 | \( 1 + (0.414 - 0.414i)T - 5iT^{2} \) |
| 7 | \( 1 - 3.69iT - 7T^{2} \) |
| 11 | \( 1 + (2.93 - 2.93i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.41 + 2.41i)T + 13iT^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + (4.46 + 4.46i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.75iT - 23T^{2} \) |
| 29 | \( 1 + (-5.24 - 5.24i)T + 29iT^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + (6.41 - 6.41i)T - 37iT^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + (-0.765 + 0.765i)T - 43iT^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 + (-3.24 + 3.24i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.765 + 0.765i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.757 + 0.757i)T + 61iT^{2} \) |
| 67 | \( 1 + (1.39 + 1.39i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.02iT - 71T^{2} \) |
| 73 | \( 1 + 6.48iT - 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + (-9.68 - 9.68i)T + 83iT^{2} \) |
| 89 | \( 1 + 4.82iT - 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−11.36759931345051465692318305576, −10.43135972958123736479830986144, −9.346176649767787981014986327307, −8.661536526080138582999097492614, −7.43027371876901485894542600584, −6.73583829467153929045766647946, −5.58249173961918211076993616101, −4.90687533677878152565787307869, −3.17065645868108135610550623055, −2.04833352019253287965533360857,
0.24354278763527971365171419129, 2.37498512763698241093886496167, 4.13351153626627673673790460807, 4.57342866015957090660895347254, 5.86364629786771182778472147411, 6.90568612379145544334886101600, 7.928497070870892205628311312865, 8.627852165704761885725463170884, 10.15764579600967817694411287149, 10.47202144242118855781509586419