L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−2.12 − 0.707i)5-s + (0.707 + 0.707i)8-s + (2 − 0.999i)10-s − 2.82i·11-s + (−3 + 3i)13-s − 1.00·16-s + (−2.82 + 2.82i)17-s − 4i·19-s + (−0.707 + 2.12i)20-s + (2.00 + 2.00i)22-s + (0.707 + 0.707i)23-s + (3.99 + 3i)25-s − 4.24i·26-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.948 − 0.316i)5-s + (0.250 + 0.250i)8-s + (0.632 − 0.316i)10-s − 0.852i·11-s + (−0.832 + 0.832i)13-s − 0.250·16-s + (−0.685 + 0.685i)17-s − 0.917i·19-s + (−0.158 + 0.474i)20-s + (0.426 + 0.426i)22-s + (0.147 + 0.147i)23-s + (0.799 + 0.600i)25-s − 0.832i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8013015653\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8013015653\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.12 + 0.707i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.82 - 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-7 - 7i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-2 + 2i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.48 + 8.48i)T + 53iT^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + (-10 - 10i)T + 67iT^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.89T + 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−9.105851463025712865115265874971, −8.356202093857425311345808977896, −7.82336593076467849371328939962, −6.90611795787722322229822338776, −6.34542991958962523680944798431, −5.16352437927969192216389878360, −4.48836831684164398197780786827, −3.52077737151280507137087883588, −2.23126284303152791430322724905, −0.73851529732837011238998420501,
0.52350747622087764878506862670, 2.15080119458826458329038494823, 2.98343336744808391698613525032, 4.03632283175555575849766188015, 4.70969786663937874919580518670, 5.87609969980516257166134290198, 7.06570378659244795528854079048, 7.56441136569945313064585197752, 8.105323519522208816773790658135, 9.180709176071292907301366047830