| L(s) = 1 | + 2·3-s − 4-s − 2·5-s − 2·7-s + 3·9-s + 2·11-s − 2·12-s − 6·13-s − 4·15-s − 3·16-s − 8·17-s + 2·20-s − 4·21-s + 2·23-s + 3·25-s + 4·27-s + 2·28-s − 6·31-s + 4·33-s + 4·35-s − 3·36-s − 6·37-s − 12·39-s − 2·44-s − 6·45-s + 4·47-s − 6·48-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s − 0.894·5-s − 0.755·7-s + 9-s + 0.603·11-s − 0.577·12-s − 1.66·13-s − 1.03·15-s − 3/4·16-s − 1.94·17-s + 0.447·20-s − 0.872·21-s + 0.417·23-s + 3/5·25-s + 0.769·27-s + 0.377·28-s − 1.07·31-s + 0.696·33-s + 0.676·35-s − 1/2·36-s − 0.986·37-s − 1.92·39-s − 0.301·44-s − 0.894·45-s + 0.583·47-s − 0.866·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5832225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5832225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 80 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 104 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 156 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 266 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.847341237474973254107692798106, −8.828646491065890938857464803399, −7.76899184368584902711427164800, −7.64316451319783038898097192388, −7.46607197750631022772281675801, −6.76510664725334241484654105232, −6.66904583219816788952858186705, −6.35667403943400842252262990263, −5.41537290257269145338776766065, −5.10800366197472759380565668438, −4.46424007057290058638831334177, −4.41922006392416609691257372630, −3.87249194634555114084418808383, −3.51555342527337066390694918364, −2.89506948493770158158820331283, −2.50862205746565734337018649490, −2.10049246126520267332129729304, −1.33644099963015153306801251782, 0, 0,
1.33644099963015153306801251782, 2.10049246126520267332129729304, 2.50862205746565734337018649490, 2.89506948493770158158820331283, 3.51555342527337066390694918364, 3.87249194634555114084418808383, 4.41922006392416609691257372630, 4.46424007057290058638831334177, 5.10800366197472759380565668438, 5.41537290257269145338776766065, 6.35667403943400842252262990263, 6.66904583219816788952858186705, 6.76510664725334241484654105232, 7.46607197750631022772281675801, 7.64316451319783038898097192388, 7.76899184368584902711427164800, 8.828646491065890938857464803399, 8.847341237474973254107692798106
