| L(s) = 1 | − 3·9-s + 10·11-s − 6·19-s + 12·29-s + 8·31-s + 22·41-s − 49-s + 8·59-s − 4·61-s + 20·71-s − 4·79-s + 22·89-s − 30·99-s + 36·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 18·171-s + 173-s + ⋯ |
| L(s) = 1 | − 9-s + 3.01·11-s − 1.37·19-s + 2.22·29-s + 1.43·31-s + 3.43·41-s − 1/7·49-s + 1.04·59-s − 0.512·61-s + 2.37·71-s − 0.450·79-s + 2.33·89-s − 3.01·99-s + 3.44·109-s + 4.81·121-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 + 1.37·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.685488049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.685488049\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 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
−9.076235760795153340601652790456, −8.680502964898596526145521334914, −8.416205533675640242505669875373, −7.85526139618735009564756144543, −7.60162871113490771013261133864, −6.85712782025260333711457713729, −6.60881873513603227130121832174, −6.35568638090845137880446584403, −6.04844473627330597333054081040, −5.81915723962884947796499165911, −4.80025853461378712326735180606, −4.77994703781332761587988288891, −4.19450755223027610189939265590, −3.83965706151057570278237292129, −3.53629591371083024295916370386, −2.66749320425673017600255589050, −2.54685890465382673269306460434, −1.81937509470877097482148290782, −0.930520821249209750120031638115, −0.849514495136806981317882134586,
0.849514495136806981317882134586, 0.930520821249209750120031638115, 1.81937509470877097482148290782, 2.54685890465382673269306460434, 2.66749320425673017600255589050, 3.53629591371083024295916370386, 3.83965706151057570278237292129, 4.19450755223027610189939265590, 4.77994703781332761587988288891, 4.80025853461378712326735180606, 5.81915723962884947796499165911, 6.04844473627330597333054081040, 6.35568638090845137880446584403, 6.60881873513603227130121832174, 6.85712782025260333711457713729, 7.60162871113490771013261133864, 7.85526139618735009564756144543, 8.416205533675640242505669875373, 8.680502964898596526145521334914, 9.076235760795153340601652790456
