L(s) = 1 | + (−1.22 + 1.22i)3-s + (2.44 + 2.44i)7-s − 2.99i·9-s + (−4.89 + 4.89i)13-s + 8i·19-s − 5.99·21-s + (3.67 + 3.67i)27-s + 4·31-s + (−4.89 − 4.89i)37-s − 11.9i·39-s + (7.34 − 7.34i)43-s + 4.99i·49-s + (−9.79 − 9.79i)57-s + 14·61-s + (7.34 − 7.34i)63-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.925 + 0.925i)7-s − 0.999i·9-s + (−1.35 + 1.35i)13-s + 1.83i·19-s − 1.30·21-s + (0.707 + 0.707i)27-s + 0.718·31-s + (−0.805 − 0.805i)37-s − 1.92i·39-s + (1.12 − 1.12i)43-s + 0.714i·49-s + (−1.29 − 1.29i)57-s + 1.79·61-s + (0.925 − 0.925i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.600575 + 0.758860i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.600575 + 0.758860i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (4.89 - 4.89i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (4.89 + 4.89i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-7.34 + 7.34i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + (-2.44 - 2.44i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-9.79 + 9.79i)T - 73iT^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (9.79 + 9.79i)T + 97iT^{2} \) |
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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
−12.10021391994180247394508129222, −11.17994107342761616850174790370, −10.13139948808572754283610991094, −9.330686424724207785598702588895, −8.353456367036923359828498620284, −7.06683226675601598557856900859, −5.82290054571889066703307141897, −5.00657973306111747453348303330, −3.97975442981480372751068180553, −2.07065228059845667830178901591,
0.799160189443931801462285114043, 2.59330614551833375819427227728, 4.62939666683130432817846544659, 5.29340370255185237460372487794, 6.76857922228371503159190183850, 7.50798738926167795514168174967, 8.271115255958706346250263979573, 9.851379432190725628770411968653, 10.75761051569832320141710226072, 11.39805598580936664178268936471