| L(s) = 1 | − 1.41i·3-s − 5-s + 3.46·7-s + 0.999·9-s + (1.73 − 2.82i)11-s − 2.44i·13-s + 1.41i·15-s − 7.34i·17-s − 3.46·19-s − 4.89i·21-s + 1.41i·23-s + 25-s − 5.65i·27-s + 4.89i·29-s + (−4.00 − 2.44i)33-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s − 0.447·5-s + 1.30·7-s + 0.333·9-s + (0.522 − 0.852i)11-s − 0.679i·13-s + 0.365i·15-s − 1.78i·17-s − 0.794·19-s − 1.06i·21-s + 0.294i·23-s + 0.200·25-s − 1.08i·27-s + 0.909i·29-s + (−0.696 − 0.426i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.012806118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.012806118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + (-1.73 + 2.82i)T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 7.34iT - 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 4.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 2.82iT - 59T^{2} \) |
| 61 | \( 1 + 9.79iT - 61T^{2} \) |
| 67 | \( 1 + 12.7iT - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 7.34iT - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.119118556114080178401223097406, −7.66184816592663156869384314262, −6.98328570396953715297523090547, −6.21962920983372476734156741630, −5.18559772208244592868642228526, −4.62399751586579729761325501141, −3.58337479966910944500072218486, −2.56673319708893730997211091270, −1.48944384754057488827269786790, −0.63996657038025462507346360366,
1.45316860875698030332886728271, 2.16503412446262740253428903473, 3.79055417514731357734420143840, 4.25790635183334551806258225916, 4.67146238988024935372452154490, 5.70808406007118106221043219455, 6.67966369062145293207164376328, 7.37293147671907900111664227073, 8.296974412460786525956555173341, 8.655645619763929732202348573754
