| L(s) = 1 | − 3-s − 5-s − 1.29·7-s + 9-s + 5.01·11-s + 3.29·13-s + 15-s + 5.71·17-s + 19-s + 1.29·21-s + 3.71·23-s + 25-s − 27-s + 4.41·29-s − 7.71·31-s − 5.01·33-s + 1.29·35-s + 3.29·37-s − 3.29·39-s − 7.01·41-s − 5.29·43-s − 45-s − 3.71·47-s − 5.31·49-s − 5.71·51-s + 5.71·53-s − 5.01·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.491·7-s + 0.333·9-s + 1.51·11-s + 0.915·13-s + 0.258·15-s + 1.38·17-s + 0.229·19-s + 0.283·21-s + 0.773·23-s + 0.200·25-s − 0.192·27-s + 0.819·29-s − 1.38·31-s − 0.872·33-s + 0.219·35-s + 0.542·37-s − 0.528·39-s − 1.09·41-s − 0.808·43-s − 0.149·45-s − 0.541·47-s − 0.758·49-s − 0.799·51-s + 0.784·53-s − 0.675·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.657904643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657904643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 1.29T + 7T^{2} \) |
| 11 | \( 1 - 5.01T + 11T^{2} \) |
| 13 | \( 1 - 3.29T + 13T^{2} \) |
| 17 | \( 1 - 5.71T + 17T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 - 4.41T + 29T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 - 3.29T + 37T^{2} \) |
| 41 | \( 1 + 7.01T + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 3.71T + 47T^{2} \) |
| 53 | \( 1 - 5.71T + 53T^{2} \) |
| 59 | \( 1 + 6.59T + 59T^{2} \) |
| 61 | \( 1 - 7.11T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 9.61T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.381855370366458709156346023690, −7.47641027159507669913966469524, −6.76375131074506409994416460273, −6.24567174371663540764457455098, −5.45471184241227539161713615691, −4.60893112676358788196324467499, −3.59554687344549631003018020592, −3.31549814423213006283430285465, −1.61760611301596332199263100631, −0.801575136536248287435852033366,
0.801575136536248287435852033366, 1.61760611301596332199263100631, 3.31549814423213006283430285465, 3.59554687344549631003018020592, 4.60893112676358788196324467499, 5.45471184241227539161713615691, 6.24567174371663540764457455098, 6.76375131074506409994416460273, 7.47641027159507669913966469524, 8.381855370366458709156346023690
