| L(s) = 1 | + (2 − i)5-s − 3i·9-s + (1 − i)13-s + (3 + 3i)17-s + (3 − 4i)25-s − 4i·29-s + (7 + 7i)37-s − 8·41-s + (−3 − 6i)45-s + 7i·49-s + (−9 + 9i)53-s − 12·61-s + (1 − 3i)65-s + (−11 + 11i)73-s − 9·81-s + ⋯ |
| L(s) = 1 | + (0.894 − 0.447i)5-s − i·9-s + (0.277 − 0.277i)13-s + (0.727 + 0.727i)17-s + (0.600 − 0.800i)25-s − 0.742i·29-s + (1.15 + 1.15i)37-s − 1.24·41-s + (−0.447 − 0.894i)45-s + i·49-s + (−1.23 + 1.23i)53-s − 1.53·61-s + (0.124 − 0.372i)65-s + (−1.28 + 1.28i)73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45848 - 0.414326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45848 - 0.414326i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| good | 3 | \( 1 + 3iT^{2} \) |
| 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3 - 3i)T + 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-7 - 7i)T + 37iT^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (9 - 9i)T - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (11 - 11i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (-13 - 13i)T + 97iT^{2} \) |
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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
−11.67884295715023212621534687179, −10.42821154254018852586134744406, −9.693419260056799415960468756177, −8.859010812383679043032619910844, −7.85329864362359783038199561841, −6.40400267991988480893327881443, −5.80780206704734417757645076831, −4.46793970751731429499311987066, −3.07029287086804376795319408489, −1.32586343342437555983892467440,
1.85772778075380890368705227763, 3.15954120480596599270267330916, 4.84141601387313376758748006505, 5.75800316685793390864568868433, 6.88035734678523391819824097259, 7.83780976762986300410096230967, 9.029930109482513194692993201510, 9.938012052388440123704117322938, 10.72213307634961733674798426066, 11.55357618751306773912596348696
