| L(s) = 1 | + 4·11-s + 8·19-s − 20·29-s − 8·31-s + 10·49-s + 28·59-s − 28·61-s + 24·71-s − 24·79-s − 24·89-s − 12·101-s − 4·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 1.83·19-s − 3.71·29-s − 1.43·31-s + 10/7·49-s + 3.64·59-s − 3.58·61-s + 2.84·71-s − 2.70·79-s − 2.54·89-s − 1.19·101-s − 0.383·109-s − 0.909·121-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 + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.969357990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.969357990\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.135621494987194465069505271885, −8.283029875137552539839404957429, −8.002696829222782484910162859193, −7.47940585717110053184105306561, −7.26920875365545365183469001472, −7.00785676840452574169044191781, −6.66116886536566051070794851904, −5.98492367549555026670610042290, −5.61321710272776980579298178670, −5.40653079398212146529516682529, −5.24981477649923373428271719079, −4.32516884805919619448121987457, −4.04105233654486842079248085501, −3.70545176466988432949125348599, −3.41189843904115168830870708221, −2.77900230776342941726089322719, −2.23174150751661651254967236708, −1.53868997527537161760754976390, −1.39474377658818006533583857835, −0.42275476545506171524972591216,
0.42275476545506171524972591216, 1.39474377658818006533583857835, 1.53868997527537161760754976390, 2.23174150751661651254967236708, 2.77900230776342941726089322719, 3.41189843904115168830870708221, 3.70545176466988432949125348599, 4.04105233654486842079248085501, 4.32516884805919619448121987457, 5.24981477649923373428271719079, 5.40653079398212146529516682529, 5.61321710272776980579298178670, 5.98492367549555026670610042290, 6.66116886536566051070794851904, 7.00785676840452574169044191781, 7.26920875365545365183469001472, 7.47940585717110053184105306561, 8.002696829222782484910162859193, 8.283029875137552539839404957429, 9.135621494987194465069505271885
