| L(s) = 1 | + 5·5-s − 8·7-s − 18·11-s + 8·13-s + 15·17-s − 23·19-s − 63·23-s + 25·25-s + 156·29-s + 85·31-s − 40·35-s + 74·37-s + 246·41-s + 190·43-s − 288·47-s − 279·49-s − 177·53-s − 90·55-s − 792·59-s − 907·61-s + 40·65-s + 322·67-s + 270·71-s + 254·73-s + 144·77-s + 1.12e3·79-s + 771·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.431·7-s − 0.493·11-s + 0.170·13-s + 0.214·17-s − 0.277·19-s − 0.571·23-s + 1/5·25-s + 0.998·29-s + 0.492·31-s − 0.193·35-s + 0.328·37-s + 0.937·41-s + 0.673·43-s − 0.893·47-s − 0.813·49-s − 0.458·53-s − 0.220·55-s − 1.74·59-s − 1.90·61-s + 0.0763·65-s + 0.587·67-s + 0.451·71-s + 0.407·73-s + 0.213·77-s + 1.59·79-s + 1.01·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 8 T + p^{3} T^{2} \) |
| 17 | \( 1 - 15 T + p^{3} T^{2} \) |
| 19 | \( 1 + 23 T + p^{3} T^{2} \) |
| 23 | \( 1 + 63 T + p^{3} T^{2} \) |
| 29 | \( 1 - 156 T + p^{3} T^{2} \) |
| 31 | \( 1 - 85 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 190 T + p^{3} T^{2} \) |
| 47 | \( 1 + 288 T + p^{3} T^{2} \) |
| 53 | \( 1 + 177 T + p^{3} T^{2} \) |
| 59 | \( 1 + 792 T + p^{3} T^{2} \) |
| 61 | \( 1 + 907 T + p^{3} T^{2} \) |
| 67 | \( 1 - 322 T + p^{3} T^{2} \) |
| 71 | \( 1 - 270 T + p^{3} T^{2} \) |
| 73 | \( 1 - 254 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1123 T + p^{3} T^{2} \) |
| 83 | \( 1 - 771 T + p^{3} T^{2} \) |
| 89 | \( 1 + 198 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1192 T + p^{3} T^{2} \) |
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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
−8.225398844805521839433516236011, −7.70976164759773662160395612660, −6.54182544911704023012729866895, −6.13517222980904073561542824236, −5.14291182623941921735475643923, −4.32306545165161775813898127044, −3.21594370819026153753836654246, −2.41966607568051469204361098211, −1.25549548530068304221365457973, 0,
1.25549548530068304221365457973, 2.41966607568051469204361098211, 3.21594370819026153753836654246, 4.32306545165161775813898127044, 5.14291182623941921735475643923, 6.13517222980904073561542824236, 6.54182544911704023012729866895, 7.70976164759773662160395612660, 8.225398844805521839433516236011
