| L(s) = 1 | − 18·5-s + 8·7-s + 36·11-s + 10·13-s − 18·17-s + 100·19-s − 72·23-s + 199·25-s − 234·29-s − 16·31-s − 144·35-s + 226·37-s − 90·41-s − 452·43-s − 432·47-s − 279·49-s + 414·53-s − 648·55-s − 684·59-s − 422·61-s − 180·65-s − 332·67-s + 360·71-s + 26·73-s + 288·77-s + 512·79-s − 1.18e3·83-s + ⋯ |
| L(s) = 1 | − 1.60·5-s + 0.431·7-s + 0.986·11-s + 0.213·13-s − 0.256·17-s + 1.20·19-s − 0.652·23-s + 1.59·25-s − 1.49·29-s − 0.0926·31-s − 0.695·35-s + 1.00·37-s − 0.342·41-s − 1.60·43-s − 1.34·47-s − 0.813·49-s + 1.07·53-s − 1.58·55-s − 1.50·59-s − 0.885·61-s − 0.343·65-s − 0.605·67-s + 0.601·71-s + 0.0416·73-s + 0.426·77-s + 0.729·79-s − 1.57·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 18 T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 16 T + p^{3} T^{2} \) |
| 37 | \( 1 - 226 T + p^{3} T^{2} \) |
| 41 | \( 1 + 90 T + p^{3} T^{2} \) |
| 43 | \( 1 + 452 T + p^{3} T^{2} \) |
| 47 | \( 1 + 432 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 684 T + p^{3} T^{2} \) |
| 61 | \( 1 + 422 T + p^{3} T^{2} \) |
| 67 | \( 1 + 332 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 - 26 T + p^{3} T^{2} \) |
| 79 | \( 1 - 512 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 630 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1054 T + p^{3} T^{2} \) |
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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
−9.818801742122374309380141857884, −8.884786456841968249860434779739, −7.979139641983915321237509317400, −7.40442090548489432782390088268, −6.35721789655447182002899734372, −5.01008536657313932803855155046, −4.03628274919766816206891011416, −3.30285783345446818870499094905, −1.45934741993750242319217522941, 0,
1.45934741993750242319217522941, 3.30285783345446818870499094905, 4.03628274919766816206891011416, 5.01008536657313932803855155046, 6.35721789655447182002899734372, 7.40442090548489432782390088268, 7.979139641983915321237509317400, 8.884786456841968249860434779739, 9.818801742122374309380141857884
