L(s) = 1 | + (−1.63 + 1.63i)2-s + (−0.722 + 0.722i)3-s − 3.35i·4-s + (0.360 + 2.20i)5-s − 2.36i·6-s + (2.14 − 2.14i)7-s + (2.22 + 2.22i)8-s + 1.95i·9-s + (−4.20 − 3.02i)10-s + (2.42 + 2.42i)12-s + (−2.96 − 2.96i)13-s + 7.02i·14-s + (−1.85 − 1.33i)15-s − 0.554·16-s + (−4.75 + 4.75i)17-s + (−3.20 − 3.20i)18-s + ⋯ |
L(s) = 1 | + (−1.15 + 1.15i)2-s + (−0.417 + 0.417i)3-s − 1.67i·4-s + (0.161 + 0.986i)5-s − 0.965i·6-s + (0.811 − 0.811i)7-s + (0.785 + 0.785i)8-s + 0.651i·9-s + (−1.32 − 0.955i)10-s + (0.700 + 0.700i)12-s + (−0.821 − 0.821i)13-s + 1.87i·14-s + (−0.479 − 0.344i)15-s − 0.138·16-s + (−1.15 + 1.15i)17-s + (−0.754 − 0.754i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.153849 - 0.209895i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.153849 - 0.209895i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.360 - 2.20i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (1.63 - 1.63i)T - 2iT^{2} \) |
| 3 | \( 1 + (0.722 - 0.722i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.14 + 2.14i)T - 7iT^{2} \) |
| 13 | \( 1 + (2.96 + 2.96i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.75 - 4.75i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.733T + 19T^{2} \) |
| 23 | \( 1 + (4.96 - 4.96i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.84T + 29T^{2} \) |
| 31 | \( 1 - 1.08T + 31T^{2} \) |
| 37 | \( 1 + (-0.0598 - 0.0598i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.36iT - 41T^{2} \) |
| 43 | \( 1 + (-1.92 - 1.92i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.22 + 7.22i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.60 + 1.60i)T - 53iT^{2} \) |
| 59 | \( 1 + 1.19iT - 59T^{2} \) |
| 61 | \( 1 - 7.37iT - 61T^{2} \) |
| 67 | \( 1 + (-3.89 - 3.89i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + (1.92 + 1.92i)T + 73iT^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + (1.21 + 1.21i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.9iT - 89T^{2} \) |
| 97 | \( 1 + (3.15 + 3.15i)T + 97iT^{2} \) |
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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
−10.78050116303231489240212515997, −10.30712927859930703828785948450, −9.657183828223594606912978981772, −8.331551272309855088981016344479, −7.69000732052999948670987924064, −7.09912685707575076069658713365, −6.02288356560841695662292865126, −5.22830507094066122358425771331, −3.91854655676399812673665680370, −1.95408349844690061304466077084,
0.21629364079977227148878091174, 1.64112167722323944342067713851, 2.48601595942807273232385384156, 4.27806514750708758573090256695, 5.30507311269602531880641121140, 6.56418326310199213680945455437, 7.79226204994779262665150540531, 8.617829344633858107944390691649, 9.313946985510247369023228994260, 9.754788794557965692497164158207