L(s) = 1 | + 2.49i·3-s + (2.11 + 0.719i)5-s + 2.92i·7-s − 3.24·9-s − 4.10·11-s + 0.0122i·13-s + (−1.79 + 5.29i)15-s + 0.155i·17-s − 4.32·19-s − 7.31·21-s + i·23-s + (3.96 + 3.04i)25-s − 0.616i·27-s + 6.79·29-s + 2.20·31-s + ⋯ |
L(s) = 1 | + 1.44i·3-s + (0.946 + 0.321i)5-s + 1.10i·7-s − 1.08·9-s − 1.23·11-s + 0.00341i·13-s + (−0.464 + 1.36i)15-s + 0.0376i·17-s − 0.992·19-s − 1.59·21-s + 0.208i·23-s + (0.792 + 0.609i)25-s − 0.118i·27-s + 1.26·29-s + 0.396·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.242952 + 1.46987i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.242952 + 1.46987i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.11 - 0.719i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 2.49iT - 3T^{2} \) |
| 7 | \( 1 - 2.92iT - 7T^{2} \) |
| 11 | \( 1 + 4.10T + 11T^{2} \) |
| 13 | \( 1 - 0.0122iT - 13T^{2} \) |
| 17 | \( 1 - 0.155iT - 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 29 | \( 1 - 6.79T + 29T^{2} \) |
| 31 | \( 1 - 2.20T + 31T^{2} \) |
| 37 | \( 1 + 4.60iT - 37T^{2} \) |
| 41 | \( 1 + 7.67T + 41T^{2} \) |
| 43 | \( 1 + 8.38iT - 43T^{2} \) |
| 47 | \( 1 - 6.38iT - 47T^{2} \) |
| 53 | \( 1 - 7.80iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 7.05T + 61T^{2} \) |
| 67 | \( 1 - 7.31iT - 67T^{2} \) |
| 71 | \( 1 + 5.84T + 71T^{2} \) |
| 73 | \( 1 + 0.727iT - 73T^{2} \) |
| 79 | \( 1 + 4.81T + 79T^{2} \) |
| 83 | \( 1 + 7.75iT - 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 - 14.8iT - 97T^{2} \) |
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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
−10.36635222404574964360594276110, −9.784223758567199143403982096155, −8.901490181150273268502562211281, −8.357914225827067329732550475102, −6.89666103412951487395758445099, −5.74665906504392104366844955800, −5.32108864829326235697813843811, −4.35237532473430386316103441843, −3.01780341038949710201133707063, −2.26024262144316486011324711554,
0.67930528228727339301741264084, 1.84838442469030958013631985726, 2.82841626576254440387751195497, 4.48919438583463313716971013628, 5.47759447193571755434214134688, 6.57690537801993171667937310722, 6.93631892298767056475268867787, 8.110915260557925140707074486440, 8.456248544729180797037192372816, 10.00632379203320298267778580088