L(s) = 1 | − 6·5-s + 9.94·7-s + 19.8·11-s − 9.94i·13-s − 19.8i·17-s − 7i·19-s + 42i·23-s + 11·25-s − 24·29-s + 39.7·31-s − 59.6·35-s − 49.7i·37-s + 39.7i·41-s − 50i·43-s + 6i·47-s + ⋯ |
L(s) = 1 | − 1.20·5-s + 1.42·7-s + 1.80·11-s − 0.765i·13-s − 1.17i·17-s − 0.368i·19-s + 1.82i·23-s + 0.440·25-s − 0.827·29-s + 1.28·31-s − 1.70·35-s − 1.34i·37-s + 0.970i·41-s − 1.16i·43-s + 0.127i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.060460984\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.060460984\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 6T + 25T^{2} \) |
| 7 | \( 1 - 9.94T + 49T^{2} \) |
| 11 | \( 1 - 19.8T + 121T^{2} \) |
| 13 | \( 1 + 9.94iT - 169T^{2} \) |
| 17 | \( 1 + 19.8iT - 289T^{2} \) |
| 19 | \( 1 + 7iT - 361T^{2} \) |
| 23 | \( 1 - 42iT - 529T^{2} \) |
| 29 | \( 1 + 24T + 841T^{2} \) |
| 31 | \( 1 - 39.7T + 961T^{2} \) |
| 37 | \( 1 + 49.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 39.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 84T + 2.80e3T^{2} \) |
| 59 | \( 1 - 19.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 89.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 53iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 60iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 119T + 5.32e3T^{2} \) |
| 79 | \( 1 - 49.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 59.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 13T + 9.40e3T^{2} \) |
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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
−9.028111907710976346468926344852, −8.058893661389358116542760926474, −7.60597730014328771094593917912, −6.88391525645532844935415827038, −5.67185538277255108298953923757, −4.78827985410799636304566297246, −4.04527932769511525663380688017, −3.24500483225515366169674681123, −1.71945944994965816961380803151, −0.68568578217185556011939731794,
1.05995840495940586714663921819, 1.98089188043623575302149695443, 3.61014978082007249744733448892, 4.26908390638120181188606431246, 4.77446407281715616483174620279, 6.26663561690423740316349028621, 6.76860970596103821388816640551, 7.953079853656083598248938597207, 8.275328887973840555744792031313, 8.994768332762005375219439241025