| L(s) = 1 | − 3-s + 4-s + 6·5-s − 7-s − 2·9-s − 12-s − 6·15-s + 16-s + 6·17-s + 6·20-s + 21-s + 17·25-s + 5·27-s − 28-s − 6·35-s − 2·36-s − 14·37-s − 2·43-s − 12·45-s − 6·47-s − 48-s − 6·49-s − 6·51-s + 12·59-s − 6·60-s + 2·63-s + 64-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 2.68·5-s − 0.377·7-s − 2/3·9-s − 0.288·12-s − 1.54·15-s + 1/4·16-s + 1.45·17-s + 1.34·20-s + 0.218·21-s + 17/5·25-s + 0.962·27-s − 0.188·28-s − 1.01·35-s − 1/3·36-s − 2.30·37-s − 0.304·43-s − 1.78·45-s − 0.875·47-s − 0.144·48-s − 6/7·49-s − 0.840·51-s + 1.56·59-s − 0.774·60-s + 0.251·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.641708916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.641708916\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.946562857758601715546974072812, −8.344527680735992356525053307896, −8.109212031498814193717580943346, −6.96449568556298499917931526210, −6.94019412385488865960361148937, −6.34998887179359330478687747079, −5.79958575424076431462182408170, −5.66417880764699758598698858020, −5.24164848449732929783448594193, −4.77536500182716535947737093418, −3.34467457020301574582723277384, −3.32295026483676787200911474950, −2.09859100780083292577684604843, −2.08829293247371856153597309217, −1.02306378019630947306317213885,
1.02306378019630947306317213885, 2.08829293247371856153597309217, 2.09859100780083292577684604843, 3.32295026483676787200911474950, 3.34467457020301574582723277384, 4.77536500182716535947737093418, 5.24164848449732929783448594193, 5.66417880764699758598698858020, 5.79958575424076431462182408170, 6.34998887179359330478687747079, 6.94019412385488865960361148937, 6.96449568556298499917931526210, 8.109212031498814193717580943346, 8.344527680735992356525053307896, 8.946562857758601715546974072812
