| L(s) = 1 | − 1.90·3-s + 5-s − 7-s + 0.634·9-s + 3.11·11-s − 0.311·13-s − 1.90·15-s + 7.60·17-s + 8.03·19-s + 1.90·21-s − 4.08·23-s + 25-s + 4.50·27-s − 2.26·29-s + 1.05·31-s − 5.92·33-s − 35-s + 5.74·37-s + 0.593·39-s − 7.37·41-s + 43-s + 0.634·45-s + 8.25·47-s + 49-s − 14.5·51-s + 6.20·53-s + 3.11·55-s + ⋯ |
| L(s) = 1 | − 1.10·3-s + 0.447·5-s − 0.377·7-s + 0.211·9-s + 0.937·11-s − 0.0864·13-s − 0.492·15-s + 1.84·17-s + 1.84·19-s + 0.416·21-s − 0.851·23-s + 0.200·25-s + 0.867·27-s − 0.420·29-s + 0.189·31-s − 1.03·33-s − 0.169·35-s + 0.944·37-s + 0.0950·39-s − 1.15·41-s + 0.152·43-s + 0.0945·45-s + 1.20·47-s + 0.142·49-s − 2.03·51-s + 0.852·53-s + 0.419·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.553993330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553993330\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 1.90T + 3T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 + 0.311T + 13T^{2} \) |
| 17 | \( 1 - 7.60T + 17T^{2} \) |
| 19 | \( 1 - 8.03T + 19T^{2} \) |
| 23 | \( 1 + 4.08T + 23T^{2} \) |
| 29 | \( 1 + 2.26T + 29T^{2} \) |
| 31 | \( 1 - 1.05T + 31T^{2} \) |
| 37 | \( 1 - 5.74T + 37T^{2} \) |
| 41 | \( 1 + 7.37T + 41T^{2} \) |
| 47 | \( 1 - 8.25T + 47T^{2} \) |
| 53 | \( 1 - 6.20T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 4.32T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 1.16T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.82493853107833804902439868750, −7.36499591274604453424951763944, −6.41740344462700599269792013602, −5.87349166688637019820508012451, −5.46428148238350193346373376502, −4.63823286879347199861433227579, −3.58605264579840754099649770693, −2.93598474003359269018173562531, −1.52153332790694336839578843989, −0.75360729375673409977095007292,
0.75360729375673409977095007292, 1.52153332790694336839578843989, 2.93598474003359269018173562531, 3.58605264579840754099649770693, 4.63823286879347199861433227579, 5.46428148238350193346373376502, 5.87349166688637019820508012451, 6.41740344462700599269792013602, 7.36499591274604453424951763944, 7.82493853107833804902439868750
