L(s) = 1 | − 1.92·2-s + 1.70·4-s + (1.22 − 1.86i)5-s + (−2.62 − 0.341i)7-s + 0.572·8-s + (−2.36 + 3.59i)10-s − 3.13i·11-s − 3.49·13-s + (5.04 + 0.657i)14-s − 4.50·16-s − 0.661i·17-s + 4.05i·19-s + (2.09 − 3.17i)20-s + 6.04i·22-s − 2.99·23-s + ⋯ |
L(s) = 1 | − 1.36·2-s + 0.851·4-s + (0.550 − 0.835i)5-s + (−0.991 − 0.129i)7-s + 0.202·8-s + (−0.748 + 1.13i)10-s − 0.946i·11-s − 0.969·13-s + (1.34 + 0.175i)14-s − 1.12·16-s − 0.160i·17-s + 0.930i·19-s + (0.468 − 0.710i)20-s + 1.28i·22-s − 0.624·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.000872788 + 0.00234746i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.000872788 + 0.00234746i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.22 + 1.86i)T \) |
| 7 | \( 1 + (2.62 + 0.341i)T \) |
good | 2 | \( 1 + 1.92T + 2T^{2} \) |
| 11 | \( 1 + 3.13iT - 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 + 0.661iT - 17T^{2} \) |
| 19 | \( 1 - 4.05iT - 19T^{2} \) |
| 23 | \( 1 + 2.99T + 23T^{2} \) |
| 29 | \( 1 + 0.604iT - 29T^{2} \) |
| 31 | \( 1 - 1.99iT - 31T^{2} \) |
| 37 | \( 1 - 8.98iT - 37T^{2} \) |
| 41 | \( 1 + 2.09T + 41T^{2} \) |
| 43 | \( 1 + 8.55iT - 43T^{2} \) |
| 47 | \( 1 - 6.12iT - 47T^{2} \) |
| 53 | \( 1 - 5.42T + 53T^{2} \) |
| 59 | \( 1 + 5.81T + 59T^{2} \) |
| 61 | \( 1 - 4.13iT - 61T^{2} \) |
| 67 | \( 1 - 10.7iT - 67T^{2} \) |
| 71 | \( 1 + 10.5iT - 71T^{2} \) |
| 73 | \( 1 - 2.64T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 14.4iT - 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−9.935690924569389522833936254017, −9.679473044703484170498433461514, −8.730288320426315734465326732846, −8.196311416326997589106343516011, −7.19489953772191704719290126112, −6.25382180585781107024911577170, −5.33047226848330364584685359713, −4.09510214189464828150023761240, −2.65980676524578256897980178280, −1.30028793585553993276452777646,
0.00189962645337687834227007593, 1.95971977666780784191838507500, 2.80731921647707140244473335809, 4.30267204538307014346200043161, 5.61669947638711256715373883427, 6.82283301233308710166691655964, 7.10499212671570037160589718652, 8.080522952651863309934426222013, 9.367530445766869967411993591415, 9.526817598244862797822776615471