| L(s) = 1 | − 2.55·5-s − 3.15·7-s + 1.04·11-s + 2.08·17-s + 2.46·19-s − 4.38·23-s + 1.52·25-s + 3.15·29-s + 4.93·31-s + 8.07·35-s + 8.56·37-s − 11.4·41-s + 12.7·43-s − 3.54·47-s + 2.97·49-s + 8.45·53-s − 2.67·55-s − 12.4·59-s + 3.57·61-s + 1.74·67-s + 4.69·71-s − 7.34·73-s − 3.31·77-s + 6.73·79-s − 2.19·83-s − 5.33·85-s − 13.1·89-s + ⋯ |
| L(s) = 1 | − 1.14·5-s − 1.19·7-s + 0.316·11-s + 0.506·17-s + 0.565·19-s − 0.914·23-s + 0.305·25-s + 0.586·29-s + 0.887·31-s + 1.36·35-s + 1.40·37-s − 1.79·41-s + 1.94·43-s − 0.516·47-s + 0.425·49-s + 1.16·53-s − 0.361·55-s − 1.62·59-s + 0.457·61-s + 0.212·67-s + 0.556·71-s − 0.859·73-s − 0.377·77-s + 0.757·79-s − 0.241·83-s − 0.578·85-s − 1.39·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.55T + 5T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 - 3.15T + 29T^{2} \) |
| 31 | \( 1 - 4.93T + 31T^{2} \) |
| 37 | \( 1 - 8.56T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 + 3.54T + 47T^{2} \) |
| 53 | \( 1 - 8.45T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 3.57T + 61T^{2} \) |
| 67 | \( 1 - 1.74T + 67T^{2} \) |
| 71 | \( 1 - 4.69T + 71T^{2} \) |
| 73 | \( 1 + 7.34T + 73T^{2} \) |
| 79 | \( 1 - 6.73T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.75319134346617770866457396460, −7.04389255301580161905483549720, −6.34385506027110206948939190438, −5.71826311092004567526947926414, −4.63461999336710654839009426674, −3.94082504791791593694857301681, −3.32618137208591313409972963104, −2.56539403013130494983121716992, −1.07466760442081363629942446789, 0,
1.07466760442081363629942446789, 2.56539403013130494983121716992, 3.32618137208591313409972963104, 3.94082504791791593694857301681, 4.63461999336710654839009426674, 5.71826311092004567526947926414, 6.34385506027110206948939190438, 7.04389255301580161905483549720, 7.75319134346617770866457396460
