| L(s) = 1 | + 5-s − 11-s + 5·13-s + 8·17-s + 5·19-s + 8·23-s + 5·25-s − 3·29-s + 2·31-s + 3·37-s − 6·41-s + 7·43-s − 12·47-s − 11·53-s − 55-s + 5·59-s + 20·61-s + 5·65-s − 7·67-s + 4·71-s − 73-s − 8·79-s + 7·83-s + 8·85-s + 6·89-s + 5·95-s + 25·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s + 1.38·13-s + 1.94·17-s + 1.14·19-s + 1.66·23-s + 25-s − 0.557·29-s + 0.359·31-s + 0.493·37-s − 0.937·41-s + 1.06·43-s − 1.75·47-s − 1.51·53-s − 0.134·55-s + 0.650·59-s + 2.56·61-s + 0.620·65-s − 0.855·67-s + 0.474·71-s − 0.117·73-s − 0.900·79-s + 0.768·83-s + 0.867·85-s + 0.635·89-s + 0.512·95-s + 2.53·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.766931500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.766931500\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 122 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 132 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + T + 132 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 164 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 130 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 25 T + 336 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
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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
−7.978972541711201537128547440650, −7.86978770103416602588758663709, −7.38432096828305060550012975114, −7.14409268122109976167282740627, −6.63162347293335634661149273941, −6.44923695614166910188326298781, −5.87111459646461488024548039744, −5.74919061410498442656330597383, −5.21640717181545932998557282690, −5.06766336960376844669487522255, −4.70738293764577312063480169105, −4.13857392682189700293555059576, −3.51135275609438919697679605344, −3.27123620528034152938493649687, −3.19439580237188933494525050552, −2.60512890550146842939370464130, −1.93561795158110418724727232009, −1.48557238731539911828864422295, −0.888183035665950526822857826380, −0.795649439728604679389783437840,
0.795649439728604679389783437840, 0.888183035665950526822857826380, 1.48557238731539911828864422295, 1.93561795158110418724727232009, 2.60512890550146842939370464130, 3.19439580237188933494525050552, 3.27123620528034152938493649687, 3.51135275609438919697679605344, 4.13857392682189700293555059576, 4.70738293764577312063480169105, 5.06766336960376844669487522255, 5.21640717181545932998557282690, 5.74919061410498442656330597383, 5.87111459646461488024548039744, 6.44923695614166910188326298781, 6.63162347293335634661149273941, 7.14409268122109976167282740627, 7.38432096828305060550012975114, 7.86978770103416602588758663709, 7.978972541711201537128547440650
