| L(s) = 1 | + 3·7-s + 11-s + 3·13-s + 8·17-s − 5·19-s − 8·29-s − 3·31-s − 6·37-s + 4·41-s + 7·43-s − 4·47-s + 2·49-s − 4·53-s − 4·59-s − 5·61-s − 13·67-s − 8·71-s + 10·73-s + 3·77-s − 4·79-s − 12·89-s + 9·91-s + 3·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.301·11-s + 0.832·13-s + 1.94·17-s − 1.14·19-s − 1.48·29-s − 0.538·31-s − 0.986·37-s + 0.624·41-s + 1.06·43-s − 0.583·47-s + 2/7·49-s − 0.549·53-s − 0.520·59-s − 0.640·61-s − 1.58·67-s − 0.949·71-s + 1.17·73-s + 0.341·77-s − 0.450·79-s − 1.27·89-s + 0.943·91-s + 0.304·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.91902490175664, −14.57171399386783, −14.10864469757201, −13.63478403238928, −12.85503825792826, −12.48544361920057, −11.93755636010797, −11.28141939932775, −10.93917116496014, −10.46437484685741, −9.772317204891558, −9.151316479198896, −8.649771099778977, −8.077131016726612, −7.586470071368334, −7.158453550748614, −6.132085909969749, −5.867976015081090, −5.184766799798246, −4.568028561004971, −3.838658383531097, −3.421927952110080, −2.477193689467145, −1.541100379365680, −1.316433233279706, 0,
1.316433233279706, 1.541100379365680, 2.477193689467145, 3.421927952110080, 3.838658383531097, 4.568028561004971, 5.184766799798246, 5.867976015081090, 6.132085909969749, 7.158453550748614, 7.586470071368334, 8.077131016726612, 8.649771099778977, 9.151316479198896, 9.772317204891558, 10.46437484685741, 10.93917116496014, 11.28141939932775, 11.93755636010797, 12.48544361920057, 12.85503825792826, 13.63478403238928, 14.10864469757201, 14.57171399386783, 14.91902490175664
