| L(s) = 1 | − 2·4-s + 7-s + 3·13-s + 4·16-s + 3·19-s + 2·25-s − 2·28-s + 16·31-s − 15·37-s + 7·49-s − 6·52-s − 15·61-s − 8·64-s + 3·67-s + 7·73-s − 6·76-s − 5·79-s + 3·91-s − 17·97-s − 4·100-s + 101-s + 103-s + 107-s + 109-s + 4·112-s + 113-s − 10·121-s + ⋯ |
| L(s) = 1 | − 4-s + 0.377·7-s + 0.832·13-s + 16-s + 0.688·19-s + 2/5·25-s − 0.377·28-s + 2.87·31-s − 2.46·37-s + 49-s − 0.832·52-s − 1.92·61-s − 64-s + 0.366·67-s + 0.819·73-s − 0.688·76-s − 0.562·79-s + 0.314·91-s − 1.72·97-s − 2/5·100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.377·112-s + 0.0940·113-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15552 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15552 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9573715901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9573715901\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 15 T + 112 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−15.8414588714, −15.5997623480, −15.1732656581, −14.4513458124, −13.9787119681, −13.5714564234, −13.5076727126, −12.5436827286, −12.1421195821, −11.8064821694, −10.9740224581, −10.5013219276, −10.0483028796, −9.41358369338, −8.84824414364, −8.34707887165, −7.99970098034, −7.14293326901, −6.47798157232, −5.73096149952, −5.08181566249, −4.49905013696, −3.71843086202, −2.89179484174, −1.27340197447,
1.27340197447, 2.89179484174, 3.71843086202, 4.49905013696, 5.08181566249, 5.73096149952, 6.47798157232, 7.14293326901, 7.99970098034, 8.34707887165, 8.84824414364, 9.41358369338, 10.0483028796, 10.5013219276, 10.9740224581, 11.8064821694, 12.1421195821, 12.5436827286, 13.5076727126, 13.5714564234, 13.9787119681, 14.4513458124, 15.1732656581, 15.5997623480, 15.8414588714
