| L(s) = 1 | − i·2-s + (−1 + i)3-s + 4-s + (1 + i)6-s + 2·7-s − 3i·8-s + i·9-s + (−1 + i)11-s + (−1 + i)12-s + (2 + 3i)13-s − 2i·14-s − 16-s + (1 − i)17-s + 18-s + (5 − 5i)19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.577 + 0.577i)3-s + 0.5·4-s + (0.408 + 0.408i)6-s + 0.755·7-s − 1.06i·8-s + 0.333i·9-s + (−0.301 + 0.301i)11-s + (−0.288 + 0.288i)12-s + (0.554 + 0.832i)13-s − 0.534i·14-s − 0.250·16-s + (0.242 − 0.242i)17-s + 0.235·18-s + (1.14 − 1.14i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39869 - 0.182557i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39869 - 0.182557i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + (-2 - 3i)T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + (1 - i)T - 11iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - 17iT^{2} \) |
| 19 | \( 1 + (-5 + 5i)T - 19iT^{2} \) |
| 23 | \( 1 + (-3 - 3i)T + 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (-5 - 5i)T + 31iT^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + (7 + 7i)T + 41iT^{2} \) |
| 43 | \( 1 + (1 + i)T + 43iT^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (5 - 5i)T - 53iT^{2} \) |
| 59 | \( 1 + (-7 - 7i)T + 59iT^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + (-1 - i)T + 71iT^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (5 + 5i)T + 89iT^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−11.49877995026866319420390841576, −10.82669726838418579362701549665, −10.05127028613271994248561309330, −9.050008011406213315356624561879, −7.67109668290368276829886412309, −6.75376157360806958794585756715, −5.35802131020787838166673963893, −4.50679979538928923128421638146, −3.05605240781393840939190521912, −1.56913327172388475004175740625,
1.38698629632226783143308340965, 3.20387542892187851939078231485, 5.10467461181155124744900307917, 5.92458878441165171090142572011, 6.68892470576552616216444542324, 7.87155574364968762775767305140, 8.253305675688649284095761861164, 9.873443212325966159410298061836, 11.01293290014164402462558776367, 11.59263846417408964018128818610
