| L(s) = 1 | + 5·9-s − 4·11-s + 12·19-s − 10·29-s + 6·31-s − 18·41-s − 2·49-s + 16·59-s − 20·61-s − 26·71-s − 12·79-s + 16·81-s + 8·89-s − 20·99-s − 28·101-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s + 60·171-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 1.20·11-s + 2.75·19-s − 1.85·29-s + 1.07·31-s − 2.81·41-s − 2/7·49-s + 2.08·59-s − 2.56·61-s − 3.08·71-s − 1.35·79-s + 16/9·81-s + 0.847·89-s − 2.01·99-s − 2.78·101-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 − 1.76·169-s + 4.58·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.179415200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.179415200\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.502013244017442453461917088328, −7.85595365004273726285780751344, −7.80085297098931541700811562981, −7.46481501934088254910356120587, −7.04238169796340768801914987062, −6.92041967769473759041771235820, −6.42591599733275193921253188435, −5.72305421419223038439549049987, −5.59319829685617699386013593843, −5.14185309033717539478456519448, −4.92624372358340004098118740065, −4.33244393723771522767141948082, −4.10278042469097302418075792641, −3.46158798904857797824759740264, −3.01758454685879549739758716312, −2.94840523912620024894492319559, −2.04004531823986470529921027543, −1.52272733420226442298179197028, −1.33110428968335554797324320085, −0.41424784018949295094134184035,
0.41424784018949295094134184035, 1.33110428968335554797324320085, 1.52272733420226442298179197028, 2.04004531823986470529921027543, 2.94840523912620024894492319559, 3.01758454685879549739758716312, 3.46158798904857797824759740264, 4.10278042469097302418075792641, 4.33244393723771522767141948082, 4.92624372358340004098118740065, 5.14185309033717539478456519448, 5.59319829685617699386013593843, 5.72305421419223038439549049987, 6.42591599733275193921253188435, 6.92041967769473759041771235820, 7.04238169796340768801914987062, 7.46481501934088254910356120587, 7.80085297098931541700811562981, 7.85595365004273726285780751344, 8.502013244017442453461917088328
