| L(s) = 1 | + 2·5-s − 4·7-s − 6·11-s + 2·13-s + 2·19-s + 6·23-s + 3·25-s + 12·29-s − 10·31-s − 8·35-s − 8·37-s − 10·43-s + 12·47-s − 2·49-s − 12·55-s − 6·59-s + 4·61-s + 4·65-s + 8·67-s + 6·71-s − 8·73-s + 24·77-s − 4·79-s − 12·83-s + 12·89-s − 8·91-s + 4·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 1.80·11-s + 0.554·13-s + 0.458·19-s + 1.25·23-s + 3/5·25-s + 2.22·29-s − 1.79·31-s − 1.35·35-s − 1.31·37-s − 1.52·43-s + 1.75·47-s − 2/7·49-s − 1.61·55-s − 0.781·59-s + 0.512·61-s + 0.496·65-s + 0.977·67-s + 0.712·71-s − 0.936·73-s + 2.73·77-s − 0.450·79-s − 1.31·83-s + 1.27·89-s − 0.838·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T - 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.47229978542791943272372106857, −7.07905586331919831142374683966, −6.71923126088275869389773117986, −6.70193363682853218740509773776, −6.12536266292652868610096126737, −5.85813292685331079696735916439, −5.38287500471936338094878781524, −5.30268893017865731197037176965, −4.81230753400074997812790481021, −4.62494796857021007946359102726, −3.82336927070056717517631096600, −3.49343301393400823453626014466, −3.18445412771892952635719943466, −2.89853780584582468983407947476, −2.34536870745855904223867008714, −2.25859159070709432565319665272, −1.21409543925472977632872485035, −1.20922237093118219500042785643, 0, 0,
1.20922237093118219500042785643, 1.21409543925472977632872485035, 2.25859159070709432565319665272, 2.34536870745855904223867008714, 2.89853780584582468983407947476, 3.18445412771892952635719943466, 3.49343301393400823453626014466, 3.82336927070056717517631096600, 4.62494796857021007946359102726, 4.81230753400074997812790481021, 5.30268893017865731197037176965, 5.38287500471936338094878781524, 5.85813292685331079696735916439, 6.12536266292652868610096126737, 6.70193363682853218740509773776, 6.71923126088275869389773117986, 7.07905586331919831142374683966, 7.47229978542791943272372106857
