| L(s) = 1 | + 2·3-s − 5-s − 2·7-s + 3·9-s − 5·11-s + 2·13-s − 2·15-s + 17-s − 5·19-s − 4·21-s − 3·23-s + 25-s + 4·27-s − 29-s − 10·33-s + 2·35-s + 15·37-s + 4·39-s − 6·41-s + 43-s − 3·45-s + 16·47-s + 3·49-s + 2·51-s − 4·53-s + 5·55-s − 10·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.755·7-s + 9-s − 1.50·11-s + 0.554·13-s − 0.516·15-s + 0.242·17-s − 1.14·19-s − 0.872·21-s − 0.625·23-s + 1/5·25-s + 0.769·27-s − 0.185·29-s − 1.74·33-s + 0.338·35-s + 2.46·37-s + 0.640·39-s − 0.937·41-s + 0.152·43-s − 0.447·45-s + 2.33·47-s + 3/7·49-s + 0.280·51-s − 0.549·53-s + 0.674·55-s − 1.32·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.903213283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.903213283\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 15 T + 120 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 132 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 156 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.475082915831101042953912842286, −8.275356686669174909933131698592, −7.73051713583992235681214913288, −7.68055848797064308405061146148, −7.22905358504426905618177421894, −6.85186836284450535489315301650, −6.29929345602442334076592727967, −6.08981399826918994713517293673, −5.67037557941313795329097237921, −5.21927172234027578866557579167, −4.53404127345482971767876496835, −4.49112396178094989313941162990, −3.82542256273838379028773987727, −3.59497070123932052884192639977, −3.16836246562289495840437359481, −2.64902807267948658255126067862, −2.17115554342392775260645601725, −2.13717117795071707278366582483, −0.990966220395811177725472490916, −0.50484284328659524175320708164,
0.50484284328659524175320708164, 0.990966220395811177725472490916, 2.13717117795071707278366582483, 2.17115554342392775260645601725, 2.64902807267948658255126067862, 3.16836246562289495840437359481, 3.59497070123932052884192639977, 3.82542256273838379028773987727, 4.49112396178094989313941162990, 4.53404127345482971767876496835, 5.21927172234027578866557579167, 5.67037557941313795329097237921, 6.08981399826918994713517293673, 6.29929345602442334076592727967, 6.85186836284450535489315301650, 7.22905358504426905618177421894, 7.68055848797064308405061146148, 7.73051713583992235681214913288, 8.275356686669174909933131698592, 8.475082915831101042953912842286
