L(s) = 1 | + i·3-s − 3.16i·5-s − 1.74·7-s − 9-s − 6.47i·11-s − 1.41i·13-s + 3.16·15-s + 2.47·17-s − 6.47i·19-s − 1.74i·21-s − 5.65·23-s − 5.00·25-s − i·27-s + 5.99i·29-s − 3.90·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 1.41i·5-s − 0.660·7-s − 0.333·9-s − 1.95i·11-s − 0.392i·13-s + 0.816·15-s + 0.599·17-s − 1.48i·19-s − 0.381i·21-s − 1.17·23-s − 1.00·25-s − 0.192i·27-s + 1.11i·29-s − 0.702·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6976957202\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6976957202\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 + 3.16iT - 5T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 + 6.47iT - 11T^{2} \) |
| 13 | \( 1 + 1.41iT - 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 5.99iT - 29T^{2} \) |
| 31 | \( 1 + 3.90T + 31T^{2} \) |
| 37 | \( 1 - 10.5iT - 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 + 1.52iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.6iT - 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 2.08iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 9.15T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 + 7.40T + 79T^{2} \) |
| 83 | \( 1 + 6.47iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.509531716477298687614340140746, −7.905832768982935243087815081357, −6.66771606306216593776238511253, −5.80589077682777862044325693898, −5.30936909889714202292474239669, −4.48977032521143098021116983329, −3.51068901669512462757998496729, −2.88513157761559430370994188929, −1.18262454012796432259325275961, −0.22602559747899664078648198914,
1.84925629362466139562175140718, 2.37151796736590827690585381302, 3.54260182105689786557728577656, 4.15782555018239913203636993585, 5.52169981088562581786623770677, 6.28362518875561054514441719608, 6.82784229746110291889110560219, 7.57690987946124431182936326606, 7.898302499050237591611998985814, 9.302942851871403501991839118496