L(s) = 1 | + 2-s − 3-s + 4-s + 0.324·5-s − 6-s − 2.43·7-s + 8-s + 9-s + 0.324·10-s + 3.40·11-s − 12-s + 0.269·13-s − 2.43·14-s − 0.324·15-s + 16-s + 4.84·17-s + 18-s + 4.02·19-s + 0.324·20-s + 2.43·21-s + 3.40·22-s − 24-s − 4.89·25-s + 0.269·26-s − 27-s − 2.43·28-s + 2.00·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.145·5-s − 0.408·6-s − 0.919·7-s + 0.353·8-s + 0.333·9-s + 0.102·10-s + 1.02·11-s − 0.288·12-s + 0.0747·13-s − 0.650·14-s − 0.0838·15-s + 0.250·16-s + 1.17·17-s + 0.235·18-s + 0.923·19-s + 0.0725·20-s + 0.530·21-s + 0.724·22-s − 0.204·24-s − 0.978·25-s + 0.0528·26-s − 0.192·27-s − 0.459·28-s + 0.373·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.465415736\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.465415736\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 0.324T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 - 3.40T + 11T^{2} \) |
| 13 | \( 1 - 0.269T + 13T^{2} \) |
| 17 | \( 1 - 4.84T + 17T^{2} \) |
| 19 | \( 1 - 4.02T + 19T^{2} \) |
| 29 | \( 1 - 2.00T + 29T^{2} \) |
| 31 | \( 1 + 9.50T + 31T^{2} \) |
| 37 | \( 1 + 2.01T + 37T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 - 3.93T + 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 + 1.08T + 53T^{2} \) |
| 59 | \( 1 - 9.90T + 59T^{2} \) |
| 61 | \( 1 - 4.35T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 1.13T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 + 1.05T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.748327395671516142516450174981, −7.59835333070444633965081173564, −7.02778435236987720476030181505, −6.23497558297678166625862892546, −5.68357406074424056606926319210, −4.98390902506183866717155665977, −3.70706040070686872532586181257, −3.51157378290755109839438703875, −2.09764865385811378511816036294, −0.901038056872679085861546948400,
0.901038056872679085861546948400, 2.09764865385811378511816036294, 3.51157378290755109839438703875, 3.70706040070686872532586181257, 4.98390902506183866717155665977, 5.68357406074424056606926319210, 6.23497558297678166625862892546, 7.02778435236987720476030181505, 7.59835333070444633965081173564, 8.748327395671516142516450174981