L(s) = 1 | + (−1 − i)5-s + (−5 + 5i)13-s − 8·17-s − 3i·25-s + (3 − 3i)29-s + (7 + 7i)37-s − 8i·41-s + 7·49-s + (9 + 9i)53-s + (11 − 11i)61-s + 10·65-s − 6i·73-s + (8 + 8i)85-s + 10i·89-s − 8·97-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.447i)5-s + (−1.38 + 1.38i)13-s − 1.94·17-s − 0.600i·25-s + (0.557 − 0.557i)29-s + (1.15 + 1.15i)37-s − 1.24i·41-s + 49-s + (1.23 + 1.23i)53-s + (1.40 − 1.40i)61-s + 1.24·65-s − 0.702i·73-s + (0.867 + 0.867i)85-s + 1.05i·89-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.167334562\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167334562\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1 + i)T + 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (5 - 5i)T - 13iT^{2} \) |
| 17 | \( 1 + 8T + 17T^{2} \) |
| 19 | \( 1 - 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (-3 + 3i)T - 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-7 - 7i)T + 37iT^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-9 - 9i)T + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (-11 + 11i)T - 61iT^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.383029175909405037583411973201, −7.48857037009772824668698172985, −6.83734430494136487379030723407, −6.27620456058626321556375541378, −5.11104595264915916438082198079, −4.41248316446448755597460755748, −4.09309636903869768195423940014, −2.59231654955138619059334936847, −2.05254973620759833811881740968, −0.51975964016386229079100706753,
0.63392393671241026761669066715, 2.26802468492041229284588118948, 2.81850783061196489178662367365, 3.83672727783151552714460959956, 4.65457869179857705037560389834, 5.36295181086017796248768573416, 6.23343047959648399832907181062, 7.18257422426829392622792156329, 7.40083421460210234008308722372, 8.385479279033368861609707424836