| L(s) = 1 | − 4-s − 3·9-s − 6·11-s + 16-s − 2·19-s − 8·29-s − 20·31-s + 3·36-s − 10·41-s + 6·44-s + 14·49-s + 8·59-s + 8·61-s − 64-s + 2·76-s − 8·79-s + 22·89-s + 18·99-s + 24·101-s + 16·109-s + 8·116-s + 5·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 9-s − 1.80·11-s + 1/4·16-s − 0.458·19-s − 1.48·29-s − 3.59·31-s + 1/2·36-s − 1.56·41-s + 0.904·44-s + 2·49-s + 1.04·59-s + 1.02·61-s − 1/8·64-s + 0.229·76-s − 0.900·79-s + 2.33·89-s + 1.80·99-s + 2.38·101-s + 1.53·109-s + 0.742·116-s + 5/11·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4834838414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4834838414\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−10.68289644424476656665108512165, −10.34305563222597825249071363127, −10.06936877878986749585616041677, −9.316346563541359854773596507880, −8.937054047937670167177999070291, −8.765309970270127063388160474161, −8.156013622075974525383855662152, −7.75129832658760489349632252111, −7.22373060948437650523184381463, −7.05880187414150937079861211622, −6.00848546992875967342402373826, −5.73412812447858352593367456974, −5.19715025946814237771748071807, −5.15326324408768038062307884309, −4.18935121509668318593566519653, −3.57324414348872353957985510826, −3.25780577683835846502962046170, −2.26825023674913546597657460255, −1.99206581273845457291176522796, −0.35568185536579578587627323658,
0.35568185536579578587627323658, 1.99206581273845457291176522796, 2.26825023674913546597657460255, 3.25780577683835846502962046170, 3.57324414348872353957985510826, 4.18935121509668318593566519653, 5.15326324408768038062307884309, 5.19715025946814237771748071807, 5.73412812447858352593367456974, 6.00848546992875967342402373826, 7.05880187414150937079861211622, 7.22373060948437650523184381463, 7.75129832658760489349632252111, 8.156013622075974525383855662152, 8.765309970270127063388160474161, 8.937054047937670167177999070291, 9.316346563541359854773596507880, 10.06936877878986749585616041677, 10.34305563222597825249071363127, 10.68289644424476656665108512165
