L(s) = 1 | − 1.34·3-s + 5-s + 2.75·7-s − 1.18·9-s + 4.31·11-s − 1.72·13-s − 1.34·15-s + 5.15·17-s + 7.66·19-s − 3.70·21-s − 8.12·23-s + 25-s + 5.63·27-s − 8.12·29-s + 6.73·31-s − 5.81·33-s + 2.75·35-s − 37-s + 2.32·39-s + 4.31·41-s + 4.32·43-s − 1.18·45-s + 4.42·47-s + 0.565·49-s − 6.95·51-s − 9.82·53-s + 4.31·55-s + ⋯ |
L(s) = 1 | − 0.778·3-s + 0.447·5-s + 1.03·7-s − 0.394·9-s + 1.30·11-s − 0.477·13-s − 0.348·15-s + 1.25·17-s + 1.75·19-s − 0.809·21-s − 1.69·23-s + 0.200·25-s + 1.08·27-s − 1.50·29-s + 1.20·31-s − 1.01·33-s + 0.464·35-s − 0.164·37-s + 0.371·39-s + 0.673·41-s + 0.660·43-s − 0.176·45-s + 0.644·47-s + 0.0807·49-s − 0.973·51-s − 1.34·53-s + 0.581·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.864685969\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.864685969\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 - 5.15T + 17T^{2} \) |
| 19 | \( 1 - 7.66T + 19T^{2} \) |
| 23 | \( 1 + 8.12T + 23T^{2} \) |
| 29 | \( 1 + 8.12T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 - 4.42T + 47T^{2} \) |
| 53 | \( 1 + 9.82T + 53T^{2} \) |
| 59 | \( 1 + 1.54T + 59T^{2} \) |
| 61 | \( 1 - 4.61T + 61T^{2} \) |
| 67 | \( 1 - 3.17T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 - 5.08T + 73T^{2} \) |
| 79 | \( 1 + 5.14T + 79T^{2} \) |
| 83 | \( 1 + 8.99T + 83T^{2} \) |
| 89 | \( 1 + 1.83T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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
−8.768896907403881859632936413693, −7.85657716428848671263125930681, −7.33900083674447209375223511160, −6.19451322990724711686833208991, −5.71733385918090655894762784449, −5.06362876927922622370074677802, −4.15288613498971970057019186059, −3.11981205762919448944044484362, −1.82673669083948028323384468685, −0.931276742715972215131005861603,
0.931276742715972215131005861603, 1.82673669083948028323384468685, 3.11981205762919448944044484362, 4.15288613498971970057019186059, 5.06362876927922622370074677802, 5.71733385918090655894762784449, 6.19451322990724711686833208991, 7.33900083674447209375223511160, 7.85657716428848671263125930681, 8.768896907403881859632936413693