| L(s) = 1 | − 2·3-s + 2·7-s + 9-s + 8·13-s − 4·21-s − 25-s + 4·27-s + 8·31-s + 12·37-s − 16·39-s + 16·43-s − 3·49-s + 16·61-s + 2·63-s + 16·67-s − 16·73-s + 2·75-s + 8·79-s − 11·81-s + 16·91-s − 16·93-s − 16·97-s + 4·103-s + 20·109-s − 24·111-s + 8·117-s − 10·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s + 2.21·13-s − 0.872·21-s − 1/5·25-s + 0.769·27-s + 1.43·31-s + 1.97·37-s − 2.56·39-s + 2.43·43-s − 3/7·49-s + 2.04·61-s + 0.251·63-s + 1.95·67-s − 1.87·73-s + 0.230·75-s + 0.900·79-s − 1.22·81-s + 1.67·91-s − 1.65·93-s − 1.62·97-s + 0.394·103-s + 1.91·109-s − 2.27·111-s + 0.739·117-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.807480327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807480327\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.302581850640388162346178815174, −7.905414662079025978817119190288, −7.49724167787234030448188108600, −6.75713333558490151738400769177, −6.36012145558933662386803684537, −6.07821294979080816053037920770, −5.66725885985225038441260145216, −5.20301284705154429031468503401, −4.62089586418568721252121177710, −4.11084204679596454995334677824, −3.77696818717280907733041953708, −2.87307071585620388790173973334, −2.29185584935972699430466549656, −1.18197020006955679560257379408, −0.893633961862432719885289291055,
0.893633961862432719885289291055, 1.18197020006955679560257379408, 2.29185584935972699430466549656, 2.87307071585620388790173973334, 3.77696818717280907733041953708, 4.11084204679596454995334677824, 4.62089586418568721252121177710, 5.20301284705154429031468503401, 5.66725885985225038441260145216, 6.07821294979080816053037920770, 6.36012145558933662386803684537, 6.75713333558490151738400769177, 7.49724167787234030448188108600, 7.905414662079025978817119190288, 8.302581850640388162346178815174
