| L(s) = 1 | − 4·11-s + 8·19-s − 4·29-s − 16·31-s − 16·41-s + 10·49-s − 12·59-s + 4·61-s − 24·71-s + 16·79-s + 24·89-s + 20·101-s − 20·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 1.83·19-s − 0.742·29-s − 2.87·31-s − 2.49·41-s + 10/7·49-s − 1.56·59-s + 0.512·61-s − 2.84·71-s + 1.80·79-s + 2.54·89-s + 1.99·101-s − 1.91·109-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−7.66555331479021233619111337319, −7.52390655197617704109886261324, −7.09399265009756006376689330319, −6.92746548779185131235980584469, −6.16181247072825141951349732501, −6.07373296723366352963268289776, −5.50345838908435585853108730863, −5.27425896519146739725991482518, −4.98425903060514589558131109936, −4.80480263141523063314349858237, −3.91935966140410246451894865106, −3.78212922331426113505408460182, −3.21000956242563228822694504276, −3.19219546727509551392310119833, −2.33362643757562758639627487293, −2.18876839971350959540882722932, −1.47631566556882664787947521864, −1.15101318083791027555382121485, 0, 0,
1.15101318083791027555382121485, 1.47631566556882664787947521864, 2.18876839971350959540882722932, 2.33362643757562758639627487293, 3.19219546727509551392310119833, 3.21000956242563228822694504276, 3.78212922331426113505408460182, 3.91935966140410246451894865106, 4.80480263141523063314349858237, 4.98425903060514589558131109936, 5.27425896519146739725991482518, 5.50345838908435585853108730863, 6.07373296723366352963268289776, 6.16181247072825141951349732501, 6.92746548779185131235980584469, 7.09399265009756006376689330319, 7.52390655197617704109886261324, 7.66555331479021233619111337319
