| L(s) = 1 | − 34·9-s + 124·13-s + 92·17-s − 180·29-s + 428·37-s − 20·41-s + 294·49-s + 1.35e3·53-s + 500·61-s − 1.04e3·73-s + 427·81-s + 1.94e3·89-s + 1.86e3·97-s − 1.20e3·101-s + 4.30e3·109-s + 4.36e3·113-s − 4.21e3·117-s − 2.58e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.12e3·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1.25·9-s + 2.64·13-s + 1.31·17-s − 1.15·29-s + 1.90·37-s − 0.0761·41-s + 6/7·49-s + 3.51·53-s + 1.04·61-s − 1.67·73-s + 0.585·81-s + 2.31·89-s + 1.95·97-s − 1.18·101-s + 3.78·109-s + 3.63·113-s − 3.33·117-s − 1.93·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 1.65·153-s + 0.000508·157-s + 0.000480·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.151945177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.151945177\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 34 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2582 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 62 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 46 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 2198 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12646 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 36462 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 154514 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 49226 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 678 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 241478 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 250 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 599106 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 581342 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 522 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 217758 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 999074 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 970 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 934 T + p^{3} T^{2} )^{2} \) |
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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
−10.12092675409220532919835194388, −9.713870029585675179267844762474, −8.909487792563950965334600984111, −8.892580047304017788719983262279, −8.497475019525728120468287910594, −8.076035840253959699687719381800, −7.47013178482860324818226382293, −7.24925138099684314018904573705, −6.39787878816246666706931039731, −6.00945740828126735454415404612, −5.68601423791891021210630987358, −5.56682118630697948194225108388, −4.65714227247657406979947700953, −4.02786195530899935473202259176, −3.51767793227875758064216706773, −3.32103381286621338463473500847, −2.49793498374524678572356945045, −1.88824811773703890918732554887, −0.840775526899223863317129440486, −0.794067071034629511385094427932,
0.794067071034629511385094427932, 0.840775526899223863317129440486, 1.88824811773703890918732554887, 2.49793498374524678572356945045, 3.32103381286621338463473500847, 3.51767793227875758064216706773, 4.02786195530899935473202259176, 4.65714227247657406979947700953, 5.56682118630697948194225108388, 5.68601423791891021210630987358, 6.00945740828126735454415404612, 6.39787878816246666706931039731, 7.24925138099684314018904573705, 7.47013178482860324818226382293, 8.076035840253959699687719381800, 8.497475019525728120468287910594, 8.892580047304017788719983262279, 8.909487792563950965334600984111, 9.713870029585675179267844762474, 10.12092675409220532919835194388
