| L(s) = 1 | − 0.480·2-s − 1.60·3-s − 1.76·4-s + 5-s + 0.770·6-s + 0.480·7-s + 1.81·8-s − 0.434·9-s − 0.480·10-s + 2.83·12-s − 4.79·13-s − 0.231·14-s − 1.60·15-s + 2.66·16-s + 2.50·17-s + 0.209·18-s − 5.75·19-s − 1.76·20-s − 0.770·21-s + 4.43·23-s − 2.90·24-s + 25-s + 2.30·26-s + 5.50·27-s − 0.850·28-s + 9.01·29-s + 0.770·30-s + ⋯ |
| L(s) = 1 | − 0.340·2-s − 0.924·3-s − 0.884·4-s + 0.447·5-s + 0.314·6-s + 0.181·7-s + 0.640·8-s − 0.144·9-s − 0.152·10-s + 0.817·12-s − 1.33·13-s − 0.0618·14-s − 0.413·15-s + 0.666·16-s + 0.606·17-s + 0.0492·18-s − 1.32·19-s − 0.395·20-s − 0.168·21-s + 0.924·23-s − 0.592·24-s + 0.200·25-s + 0.452·26-s + 1.05·27-s − 0.160·28-s + 1.67·29-s + 0.140·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6727341839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6727341839\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.480T + 2T^{2} \) |
| 3 | \( 1 + 1.60T + 3T^{2} \) |
| 7 | \( 1 - 0.480T + 7T^{2} \) |
| 13 | \( 1 + 4.79T + 13T^{2} \) |
| 17 | \( 1 - 2.50T + 17T^{2} \) |
| 19 | \( 1 + 5.75T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 - 9.01T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 - 5.20T + 37T^{2} \) |
| 41 | \( 1 + 8.45T + 41T^{2} \) |
| 43 | \( 1 - 8.53T + 43T^{2} \) |
| 47 | \( 1 - 9.60T + 47T^{2} \) |
| 53 | \( 1 - 6.10T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 - 4.02T + 61T^{2} \) |
| 67 | \( 1 - 7.60T + 67T^{2} \) |
| 71 | \( 1 + 2.76T + 71T^{2} \) |
| 73 | \( 1 + 5.00T + 73T^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 + 1.71T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 3.63T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.34711143611960531340954282910, −10.05058259454849803616164164224, −8.899275392054129501120735781151, −8.204937509568556490294971733653, −7.02136049578876090603084874180, −6.01147802039049128669735219462, −5.05274589018104801506303290634, −4.46280926403351647467884141200, −2.67245703990020557550375306389, −0.794318638946469023667388282742,
0.794318638946469023667388282742, 2.67245703990020557550375306389, 4.46280926403351647467884141200, 5.05274589018104801506303290634, 6.01147802039049128669735219462, 7.02136049578876090603084874180, 8.204937509568556490294971733653, 8.899275392054129501120735781151, 10.05058259454849803616164164224, 10.34711143611960531340954282910
