| L(s) = 1 | − 5-s − 7-s + 7·11-s − 2·13-s − 17-s − 2·19-s + 10·23-s − 5·25-s + 4·31-s + 35-s − 16·37-s + 2·41-s − 43-s − 47-s − 9·49-s + 8·53-s − 7·55-s + 24·59-s + 61-s + 2·65-s + 4·67-s − 5·73-s − 7·77-s + 2·79-s + 8·83-s + 85-s + 12·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 2.11·11-s − 0.554·13-s − 0.242·17-s − 0.458·19-s + 2.08·23-s − 25-s + 0.718·31-s + 0.169·35-s − 2.63·37-s + 0.312·41-s − 0.152·43-s − 0.145·47-s − 9/7·49-s + 1.09·53-s − 0.943·55-s + 3.12·59-s + 0.128·61-s + 0.248·65-s + 0.488·67-s − 0.585·73-s − 0.797·77-s + 0.225·79-s + 0.878·83-s + 0.108·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.293822498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.293822498\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 56 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 5 T - 56 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.811271956755484105735883188250, −8.780530311481454533284308407795, −8.407125021088330010015534185003, −7.946844000161093680133625180627, −7.29230838927439506773497904809, −6.99774550723504708137862749777, −6.78861536407159452474600459067, −6.61853467458551767799179141010, −5.87766485766901116335377328835, −5.67692009011311142564267404269, −4.92451338795402021498930386138, −4.79891104823927687026714235735, −4.20090984279139781673494724554, −3.77034792848932059167411299865, −3.42816083367264814267277313218, −3.11114301211478237098068286625, −2.21162466423133495995436614675, −1.92426994900820057172676529012, −1.12267662912189207278137653634, −0.55791708194519210209535781600,
0.55791708194519210209535781600, 1.12267662912189207278137653634, 1.92426994900820057172676529012, 2.21162466423133495995436614675, 3.11114301211478237098068286625, 3.42816083367264814267277313218, 3.77034792848932059167411299865, 4.20090984279139781673494724554, 4.79891104823927687026714235735, 4.92451338795402021498930386138, 5.67692009011311142564267404269, 5.87766485766901116335377328835, 6.61853467458551767799179141010, 6.78861536407159452474600459067, 6.99774550723504708137862749777, 7.29230838927439506773497904809, 7.946844000161093680133625180627, 8.407125021088330010015534185003, 8.780530311481454533284308407795, 8.811271956755484105735883188250
