L(s) = 1 | + 2.38i·2-s + (−0.648 − 0.648i)3-s − 3.67·4-s + (1.54 − 1.54i)6-s − 2.38·7-s − 3.99i·8-s − 2.15i·9-s + (−3.88 − 3.88i)11-s + (2.38 + 2.38i)12-s + (−2.76 + 2.31i)13-s − 5.67i·14-s + 2.15·16-s + (−0.262 − 0.262i)17-s + 5.14·18-s + (0.587 + 0.587i)19-s + ⋯ |
L(s) = 1 | + 1.68i·2-s + (−0.374 − 0.374i)3-s − 1.83·4-s + (0.630 − 0.630i)6-s − 0.900·7-s − 1.41i·8-s − 0.719i·9-s + (−1.17 − 1.17i)11-s + (0.687 + 0.687i)12-s + (−0.767 + 0.640i)13-s − 1.51i·14-s + 0.539·16-s + (−0.0637 − 0.0637i)17-s + 1.21·18-s + (0.134 + 0.134i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.115436 - 0.0820068i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.115436 - 0.0820068i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (2.76 - 2.31i)T \) |
good | 2 | \( 1 - 2.38iT - 2T^{2} \) |
| 3 | \( 1 + (0.648 + 0.648i)T + 3iT^{2} \) |
| 7 | \( 1 + 2.38T + 7T^{2} \) |
| 11 | \( 1 + (3.88 + 3.88i)T + 11iT^{2} \) |
| 17 | \( 1 + (0.262 + 0.262i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.587 - 0.587i)T + 19iT^{2} \) |
| 23 | \( 1 + (-1.98 + 1.98i)T - 23iT^{2} \) |
| 29 | \( 1 - 7.92iT - 29T^{2} \) |
| 31 | \( 1 + (-6.59 + 6.59i)T - 31iT^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + (6.37 - 6.37i)T - 41iT^{2} \) |
| 43 | \( 1 + (-3.09 + 3.09i)T - 43iT^{2} \) |
| 47 | \( 1 + 5.67T + 47T^{2} \) |
| 53 | \( 1 + (-1.54 - 1.54i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.49 - 5.49i)T - 59iT^{2} \) |
| 61 | \( 1 - 4.11T + 61T^{2} \) |
| 67 | \( 1 + 3.74iT - 67T^{2} \) |
| 71 | \( 1 + (2.04 - 2.04i)T - 71iT^{2} \) |
| 73 | \( 1 + 7.56iT - 73T^{2} \) |
| 79 | \( 1 + 6.07iT - 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + (-2.23 + 2.23i)T - 89iT^{2} \) |
| 97 | \( 1 - 1.88iT - 97T^{2} \) |
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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
−11.57501863217132959448153525706, −10.23753611071187300507301467048, −9.199187286442710626469758685931, −8.397312818247469682996128778516, −7.28720709553042343935368293957, −6.58254372554846991156255147337, −5.83701697191244495828814274897, −4.85681546942760525330045832233, −3.21157683174787472917781739898, −0.099305387926241839635564063081,
2.18823558191786331387893412864, 3.16295269506788161285369252656, 4.59688139032288010922313984845, 5.30058118984371356683309833646, 7.07443022090989052771686610028, 8.297979578988262820331734505200, 9.749647834738016582507203202527, 10.06158704082668208925982626435, 10.69789155009804562587835585040, 11.77345755251639954441316676941