L(s) = 1 | + (2.12 − 2.12i)3-s + (−2.24 − 2.24i)5-s + 9.00i·7-s − 8.99i·9-s + (−11.0 − 11.0i)11-s + (−54.5 + 54.5i)13-s − 9.51·15-s + 44.0·17-s + (−49.9 + 49.9i)19-s + (19.0 + 19.0i)21-s + 117. i·23-s − 114. i·25-s + (−19.0 − 19.0i)27-s + (−40.6 + 40.6i)29-s − 196.·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.200 − 0.200i)5-s + 0.486i·7-s − 0.333i·9-s + (−0.302 − 0.302i)11-s + (−1.16 + 1.16i)13-s − 0.163·15-s + 0.627·17-s + (−0.603 + 0.603i)19-s + (0.198 + 0.198i)21-s + 1.06i·23-s − 0.919i·25-s + (−0.136 − 0.136i)27-s + (−0.260 + 0.260i)29-s − 1.13·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9128432492\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9128432492\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.12 + 2.12i)T \) |
good | 5 | \( 1 + (2.24 + 2.24i)T + 125iT^{2} \) |
| 7 | \( 1 - 9.00iT - 343T^{2} \) |
| 11 | \( 1 + (11.0 + 11.0i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (54.5 - 54.5i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 - 44.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + (49.9 - 49.9i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 - 117. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (40.6 - 40.6i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-248. - 248. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 457. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (204. + 204. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-138. - 138. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-263. - 263. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (29.1 - 29.1i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (508. - 508. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 788. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 92.2iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 174.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (914. - 914. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.45e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 229.T + 9.12e5T^{2} \) |
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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
−11.47241302683775313843366594191, −10.08771641960912050932831014386, −9.333088749081003215537117232058, −8.357203000420221318378559433206, −7.57036201675579059541412124842, −6.53570596011674388381254534076, −5.40002429755667928083245410163, −4.19633154515306596627533874980, −2.84033726934747738944571490426, −1.63394361288582832367447651800,
0.27921396454470181726254629150, 2.32630709032737919219807372306, 3.45340228912677768150857744163, 4.62717172854035003446759516965, 5.61023959420818020339953047862, 7.15833255187075385027366365308, 7.71245952948891303792298206923, 8.818096928800954156734552192665, 9.879537567268274443195612299153, 10.47903717454697893287408867330