| L(s) = 1 | + 2-s − 3-s + 6·5-s − 6-s − 7-s − 8-s + 3·9-s + 6·10-s + 5·13-s − 14-s − 6·15-s − 16-s − 6·17-s + 3·18-s + 4·19-s + 21-s − 3·23-s + 24-s + 17·25-s + 5·26-s − 8·27-s − 6·29-s − 6·30-s − 20·31-s − 6·34-s − 6·35-s − 8·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 2.68·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 9-s + 1.89·10-s + 1.38·13-s − 0.267·14-s − 1.54·15-s − 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.917·19-s + 0.218·21-s − 0.625·23-s + 0.204·24-s + 17/5·25-s + 0.980·26-s − 1.53·27-s − 1.11·29-s − 1.09·30-s − 3.59·31-s − 1.02·34-s − 1.01·35-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.166135346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.166135346\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−13.07143906908965045159210577498, −12.73922160353387146198434040082, −12.02190406365866744120510453941, −11.38020100704506347292516012666, −10.78721891217236784158481736057, −10.49884269126765553696053207954, −9.900593717553013924300779274979, −9.456902189546710654541847457288, −8.960686021797229465600084173215, −8.805927649161906188459762946926, −7.23504235909376310207662266414, −7.17735313425718625349815130943, −6.25499851805344681387579327390, −5.76171255776173865936453051182, −5.68673696003673305533051730331, −5.08981816779652843697140847486, −4.00167907197545543142819100346, −3.53039362145739079262625116401, −1.97671970345519382583485730532, −1.85259790637911104631156538219,
1.85259790637911104631156538219, 1.97671970345519382583485730532, 3.53039362145739079262625116401, 4.00167907197545543142819100346, 5.08981816779652843697140847486, 5.68673696003673305533051730331, 5.76171255776173865936453051182, 6.25499851805344681387579327390, 7.17735313425718625349815130943, 7.23504235909376310207662266414, 8.805927649161906188459762946926, 8.960686021797229465600084173215, 9.456902189546710654541847457288, 9.900593717553013924300779274979, 10.49884269126765553696053207954, 10.78721891217236784158481736057, 11.38020100704506347292516012666, 12.02190406365866744120510453941, 12.73922160353387146198434040082, 13.07143906908965045159210577498
