| L(s) = 1 | + (−2 + i)5-s + 3i·9-s + (5 + 5i)13-s + (−5 + 5i)17-s + (3 − 4i)25-s − 4i·29-s + (−5 + 5i)37-s + 8·41-s + (−3 − 6i)45-s − 7i·49-s + (−5 − 5i)53-s + 12·61-s + (−15 − 5i)65-s + (5 + 5i)73-s − 9·81-s + ⋯ |
| L(s) = 1 | + (−0.894 + 0.447i)5-s + i·9-s + (1.38 + 1.38i)13-s + (−1.21 + 1.21i)17-s + (0.600 − 0.800i)25-s − 0.742i·29-s + (−0.821 + 0.821i)37-s + 1.24·41-s + (−0.447 − 0.894i)45-s − i·49-s + (−0.686 − 0.686i)53-s + 1.53·61-s + (−1.86 − 0.620i)65-s + (0.585 + 0.585i)73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.748467 + 0.684014i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.748467 + 0.684014i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-5 - 5i)T + 13iT^{2} \) |
| 17 | \( 1 + (5 - 5i)T - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-5 - 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (-5 + 5i)T - 97iT^{2} \) |
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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
−11.45823637494045206387330414327, −11.17739061108679690381095781286, −10.21521912329344702226891179652, −8.753983348100711241549479155291, −8.203325745555624988059499181647, −7.00036321406455903231163002005, −6.17080292838263670808391925213, −4.55279357045582279894781279974, −3.73628229388653838275156087841, −2.02547983030528401194618204137,
0.76085770125354409810161120625, 3.12558842572288379734781975579, 4.12396539936214544141183389183, 5.40503843578542021326014796610, 6.59336922864393581770584593663, 7.65282297564612043595752202219, 8.674534544629272867030782860446, 9.290739722645913941443564781581, 10.75088767626022843866068533848, 11.34802083647064143476429057640
