L(s) = 1 | + (0.587 + 0.809i)3-s + (0.892 − 2.05i)5-s − 4.13i·7-s + (−0.309 + 0.951i)9-s + (−1.16 − 3.58i)11-s + (0.664 + 0.215i)13-s + (2.18 − 0.482i)15-s + (−3.11 + 4.28i)17-s + (4.63 + 3.37i)19-s + (3.34 − 2.42i)21-s + (5.19 − 1.68i)23-s + (−3.40 − 3.66i)25-s + (−0.951 + 0.309i)27-s + (−5.68 + 4.12i)29-s + (8.16 + 5.93i)31-s + ⋯ |
L(s) = 1 | + (0.339 + 0.467i)3-s + (0.399 − 0.916i)5-s − 1.56i·7-s + (−0.103 + 0.317i)9-s + (−0.350 − 1.08i)11-s + (0.184 + 0.0598i)13-s + (0.563 − 0.124i)15-s + (−0.754 + 1.03i)17-s + (1.06 + 0.773i)19-s + (0.729 − 0.530i)21-s + (1.08 − 0.352i)23-s + (−0.681 − 0.732i)25-s + (−0.183 + 0.0594i)27-s + (−1.05 + 0.766i)29-s + (1.46 + 1.06i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37748 - 0.549149i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37748 - 0.549149i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (-0.892 + 2.05i)T \) |
good | 7 | \( 1 + 4.13iT - 7T^{2} \) |
| 11 | \( 1 + (1.16 + 3.58i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.664 - 0.215i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (3.11 - 4.28i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.63 - 3.37i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-5.19 + 1.68i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (5.68 - 4.12i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-8.16 - 5.93i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-5.50 - 1.78i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.03 - 6.27i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 4.79iT - 43T^{2} \) |
| 47 | \( 1 + (5.68 + 7.82i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.99 + 2.74i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.230 - 0.708i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.64 - 11.2i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (2.81 - 3.88i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-2.54 + 1.84i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-10.2 + 3.32i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (7.38 - 5.36i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (4.96 - 6.82i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.04 - 3.22i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (6.13 + 8.44i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.41816327209445857421755566974, −10.55599526123296328780665222379, −9.862697422305283593978686451288, −8.699609200946904874459010603574, −8.061245709256135432254906790392, −6.75207276074516998305883954273, −5.45288998164262372941114547294, −4.37711248611742869223028912357, −3.34258410744677071103678213648, −1.19709850597878694792946333474,
2.24092502285280655608692705374, 2.91267675505473961953426885759, 4.92297976372415775440930799829, 6.02513680741255936708925659548, 7.02072318742235492384038808931, 7.891645717089069537635658725139, 9.360814555300916045504483722605, 9.548952442792454249781876522927, 11.20680718693125717245846688287, 11.72211376960276837251327227442