| L(s) = 1 | − 2·7-s + 11-s − 13-s + 17-s + 4·19-s − 23-s − 5·29-s + 31-s − 6·37-s − 7·43-s − 7·47-s − 3·49-s + 12·53-s − 4·59-s + 10·61-s + 4·67-s + 12·71-s − 6·73-s − 2·77-s + 15·79-s − 2·83-s − 12·89-s + 2·91-s − 10·97-s − 15·101-s + 14·103-s − 18·107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s − 0.277·13-s + 0.242·17-s + 0.917·19-s − 0.208·23-s − 0.928·29-s + 0.179·31-s − 0.986·37-s − 1.06·43-s − 1.02·47-s − 3/7·49-s + 1.64·53-s − 0.520·59-s + 1.28·61-s + 0.488·67-s + 1.42·71-s − 0.702·73-s − 0.227·77-s + 1.68·79-s − 0.219·83-s − 1.27·89-s + 0.209·91-s − 1.01·97-s − 1.49·101-s + 1.37·103-s − 1.74·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82345535627533775783630350642, −6.96719853107478672408551150634, −6.55542449257964893880718392929, −5.55740680485512638999219336582, −5.06535194220715717625815263592, −3.89874794188658379093535487826, −3.37995510842951450719705981465, −2.42314902571864051813800069073, −1.32227596152825304196870381381, 0,
1.32227596152825304196870381381, 2.42314902571864051813800069073, 3.37995510842951450719705981465, 3.89874794188658379093535487826, 5.06535194220715717625815263592, 5.55740680485512638999219336582, 6.55542449257964893880718392929, 6.96719853107478672408551150634, 7.82345535627533775783630350642
