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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 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 | \( 1 + 14.3T + 97T^{2} \) |
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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
−9.248329402725520298627827290488, −8.310790458856988685857869035255, −7.60954227900283062605951550969, −6.71668151187807551187618104622, −5.73412547357783529378775626700, −5.06369060122924592730324392741, −3.59851549517809955264958061801, −2.92960917234787921448951887194, −2.06256289697580444860918344246, 0,
2.06256289697580444860918344246, 2.92960917234787921448951887194, 3.59851549517809955264958061801, 5.06369060122924592730324392741, 5.73412547357783529378775626700, 6.71668151187807551187618104622, 7.60954227900283062605951550969, 8.310790458856988685857869035255, 9.248329402725520298627827290488