| L(s) = 1 | − 5-s − 11-s − 2·13-s − 6·17-s + 4·19-s + 25-s + 6·29-s − 8·31-s − 10·37-s + 6·41-s + 8·43-s + 6·53-s + 55-s + 6·59-s + 4·61-s + 2·65-s + 14·67-s − 2·73-s − 10·79-s − 6·83-s + 6·85-s − 18·89-s − 4·95-s − 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 1.64·37-s + 0.937·41-s + 1.21·43-s + 0.824·53-s + 0.134·55-s + 0.781·59-s + 0.512·61-s + 0.248·65-s + 1.71·67-s − 0.234·73-s − 1.12·79-s − 0.658·83-s + 0.650·85-s − 1.90·89-s − 0.410·95-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.98084310368791, −13.66142765418573, −12.88364619537373, −12.63174826585797, −12.12796885127564, −11.48789575886403, −11.15299933429060, −10.66275354239947, −10.08787914087834, −9.585482240540665, −8.988268055113452, −8.581739958961517, −8.086843836059917, −7.278883359599139, −7.133750163663593, −6.593999087580584, −5.717061322265028, −5.351580948503337, −4.744334218562428, −4.123037253251747, −3.698456133279683, −2.826243446505902, −2.435867197971925, −1.663218836115853, −0.7595654701738363, 0,
0.7595654701738363, 1.663218836115853, 2.435867197971925, 2.826243446505902, 3.698456133279683, 4.123037253251747, 4.744334218562428, 5.351580948503337, 5.717061322265028, 6.593999087580584, 7.133750163663593, 7.278883359599139, 8.086843836059917, 8.581739958961517, 8.988268055113452, 9.585482240540665, 10.08787914087834, 10.66275354239947, 11.15299933429060, 11.48789575886403, 12.12796885127564, 12.63174826585797, 12.88364619537373, 13.66142765418573, 13.98084310368791
