| L(s) = 1 | − 3·7-s + 6·11-s − 2·13-s + 2·17-s − 4·23-s + 3·29-s + 2·31-s − 6·37-s + 6·41-s + 43-s + 6·47-s + 2·49-s + 11·53-s + 5·59-s − 10·61-s + 67-s + 8·71-s − 16·73-s − 18·77-s + 2·79-s − 89-s + 6·91-s − 13·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 1.80·11-s − 0.554·13-s + 0.485·17-s − 0.834·23-s + 0.557·29-s + 0.359·31-s − 0.986·37-s + 0.937·41-s + 0.152·43-s + 0.875·47-s + 2/7·49-s + 1.51·53-s + 0.650·59-s − 1.28·61-s + 0.122·67-s + 0.949·71-s − 1.87·73-s − 2.05·77-s + 0.225·79-s − 0.105·89-s + 0.628·91-s − 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.04406590928270, −12.50632162409406, −12.19532594974711, −11.86174348689114, −11.44084499023855, −10.66957229980189, −10.27321838888211, −9.813115193947320, −9.409196227015785, −9.028119577219435, −8.535305695872012, −7.974270490903031, −7.203871217729581, −7.029435241259023, −6.448337723547538, −5.987520312614059, −5.648205907326715, −4.841590504851659, −4.218141440508630, −3.868791744184344, −3.350785539339780, −2.747712533252890, −2.156882457405975, −1.375590119686857, −0.8191139809940427, 0,
0.8191139809940427, 1.375590119686857, 2.156882457405975, 2.747712533252890, 3.350785539339780, 3.868791744184344, 4.218141440508630, 4.841590504851659, 5.648205907326715, 5.987520312614059, 6.448337723547538, 7.029435241259023, 7.203871217729581, 7.974270490903031, 8.535305695872012, 9.028119577219435, 9.409196227015785, 9.813115193947320, 10.27321838888211, 10.66957229980189, 11.44084499023855, 11.86174348689114, 12.19532594974711, 12.50632162409406, 13.04406590928270
