| L(s) = 1 | − 2·3-s − 4-s + 4·5-s + 3·9-s − 5·11-s + 2·12-s − 8·15-s + 16-s − 4·20-s − 8·23-s + 3·25-s − 4·27-s + 10·33-s − 3·36-s − 8·37-s + 5·44-s + 12·45-s − 8·47-s − 2·48-s − 10·49-s + 4·53-s − 20·55-s + 8·60-s − 64-s + 10·67-s + 16·69-s − 16·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1.78·5-s + 9-s − 1.50·11-s + 0.577·12-s − 2.06·15-s + 1/4·16-s − 0.894·20-s − 1.66·23-s + 3/5·25-s − 0.769·27-s + 1.74·33-s − 1/2·36-s − 1.31·37-s + 0.753·44-s + 1.78·45-s − 1.16·47-s − 0.288·48-s − 1.42·49-s + 0.549·53-s − 2.69·55-s + 1.03·60-s − 1/8·64-s + 1.22·67-s + 1.92·69-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4186116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4186116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5209155048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5209155048\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 107 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
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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
−7.30591076700512291207229453181, −6.88688743030249869248946032004, −6.36785082564243195136601871292, −5.96224129603566843086955912802, −5.85571370009183196351845908600, −5.34566940026696345461148530399, −5.07882254324723824164343214188, −4.72157761729929309074516796399, −4.12561958631602396561232649159, −3.57988004623828222429462993347, −2.95509890264131384133242687425, −2.26665128840135131562562069362, −1.88405289075638571829544397190, −1.40682591986507233879597151046, −0.27114928433307480098999384190,
0.27114928433307480098999384190, 1.40682591986507233879597151046, 1.88405289075638571829544397190, 2.26665128840135131562562069362, 2.95509890264131384133242687425, 3.57988004623828222429462993347, 4.12561958631602396561232649159, 4.72157761729929309074516796399, 5.07882254324723824164343214188, 5.34566940026696345461148530399, 5.85571370009183196351845908600, 5.96224129603566843086955912802, 6.36785082564243195136601871292, 6.88688743030249869248946032004, 7.30591076700512291207229453181
