| L(s) = 1 | − 8·11-s − 2·19-s + 8·29-s − 10·31-s − 24·41-s + 13·49-s + 16·59-s + 14·61-s − 24·71-s − 24·79-s − 24·101-s + 14·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 0.458·19-s + 1.48·29-s − 1.79·31-s − 3.74·41-s + 13/7·49-s + 2.08·59-s + 1.79·61-s − 2.84·71-s − 2.70·79-s − 2.38·101-s + 1.34·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5880011866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5880011866\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−9.624594150117071153702694186486, −8.760109176557023419014752144730, −8.600888289893602811029976215309, −8.481755574450981704992289254749, −7.970449111694630378196261404925, −7.31484494826072876065514097134, −7.27564377541195974041818948170, −6.80994110932721019354073586753, −6.30409406156215446806246190472, −5.57165962929901925009274085807, −5.47849298858040102380394019960, −5.11505153136689768039395941111, −4.65685115078120863180217888785, −4.02185097798549277711437380342, −3.62327964654925743182584638950, −2.86750209863824178041767569355, −2.68840917913883505170955410335, −2.06441815840366223132747920604, −1.41375401408698772125564247912, −0.27801515378856900081906100896,
0.27801515378856900081906100896, 1.41375401408698772125564247912, 2.06441815840366223132747920604, 2.68840917913883505170955410335, 2.86750209863824178041767569355, 3.62327964654925743182584638950, 4.02185097798549277711437380342, 4.65685115078120863180217888785, 5.11505153136689768039395941111, 5.47849298858040102380394019960, 5.57165962929901925009274085807, 6.30409406156215446806246190472, 6.80994110932721019354073586753, 7.27564377541195974041818948170, 7.31484494826072876065514097134, 7.970449111694630378196261404925, 8.481755574450981704992289254749, 8.600888289893602811029976215309, 8.760109176557023419014752144730, 9.624594150117071153702694186486
