| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s + 2·13-s − 4·15-s + 4·17-s + 8·19-s − 8·23-s + 3·25-s + 4·27-s − 4·29-s + 8·31-s − 4·37-s + 4·39-s − 4·41-s + 8·43-s − 6·45-s + 8·47-s − 6·49-s + 8·51-s − 4·53-s + 16·57-s + 12·61-s − 4·65-s − 16·69-s − 16·71-s + 20·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s + 0.554·13-s − 1.03·15-s + 0.970·17-s + 1.83·19-s − 1.66·23-s + 3/5·25-s + 0.769·27-s − 0.742·29-s + 1.43·31-s − 0.657·37-s + 0.640·39-s − 0.624·41-s + 1.21·43-s − 0.894·45-s + 1.16·47-s − 6/7·49-s + 1.12·51-s − 0.549·53-s + 2.11·57-s + 1.53·61-s − 0.496·65-s − 1.92·69-s − 1.89·71-s + 2.34·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.596110127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.596110127\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_4$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 174 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.999009734022219590190478181540, −7.986670566002030683395853942275, −7.58357159965533610354586005159, −7.45470429801986328900002231543, −6.81880733544531876910030350907, −6.69353084831137070819910273190, −6.02336281164342769701599003632, −5.81837700760266866724089808489, −5.17551722681849086834980266010, −5.14687088372028335536484851135, −4.33294874046153155615714486842, −4.18758962183782718635967707456, −3.63002299131087803642245995445, −3.56283926799386805248613475986, −2.96053767648405298717827732619, −2.78880310297802647189884345659, −2.08485894192586346884349811534, −1.63672031196530829095024112529, −1.05850637172645494585055398586, −0.56492743521116634883363645384,
0.56492743521116634883363645384, 1.05850637172645494585055398586, 1.63672031196530829095024112529, 2.08485894192586346884349811534, 2.78880310297802647189884345659, 2.96053767648405298717827732619, 3.56283926799386805248613475986, 3.63002299131087803642245995445, 4.18758962183782718635967707456, 4.33294874046153155615714486842, 5.14687088372028335536484851135, 5.17551722681849086834980266010, 5.81837700760266866724089808489, 6.02336281164342769701599003632, 6.69353084831137070819910273190, 6.81880733544531876910030350907, 7.45470429801986328900002231543, 7.58357159965533610354586005159, 7.986670566002030683395853942275, 7.999009734022219590190478181540
