| L(s) = 1 | + 2·2-s + 3·4-s + 2·7-s + 4·8-s + 4·11-s + 2·13-s + 4·14-s + 5·16-s + 2·17-s + 6·19-s + 8·22-s − 4·23-s + 4·26-s + 6·28-s + 10·29-s + 6·32-s + 4·34-s − 4·37-s + 12·38-s + 8·41-s + 12·44-s − 8·46-s − 2·47-s + 3·49-s + 6·52-s + 6·53-s + 8·56-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.755·7-s + 1.41·8-s + 1.20·11-s + 0.554·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s + 1.37·19-s + 1.70·22-s − 0.834·23-s + 0.784·26-s + 1.13·28-s + 1.85·29-s + 1.06·32-s + 0.685·34-s − 0.657·37-s + 1.94·38-s + 1.24·41-s + 1.80·44-s − 1.17·46-s − 0.291·47-s + 3/7·49-s + 0.832·52-s + 0.824·53-s + 1.06·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.33545880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.33545880\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 73 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 105 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 161 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 93 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 156 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 108 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.63262370111043843131704370241, −7.57029916872560515994074720650, −7.01428659592894796935823290869, −6.80666350852720939488415435772, −6.44570425849597626422251060666, −6.02262762301723452008359783956, −5.79160217109100923462112886668, −5.52296296149824219893407750670, −4.93463182735863957646733011930, −4.87661083685185536396222788999, −4.27055051262382691251328816138, −4.16643633425528737547236867545, −3.58733371813895854385652933655, −3.46198825228872491364336373573, −2.90431789479753584336146968192, −2.57750537949978826207225020711, −1.97799695347965926190689532453, −1.61730764617821553447878227881, −1.04477543778868374438900366271, −0.806535013713941230570236808319,
0.806535013713941230570236808319, 1.04477543778868374438900366271, 1.61730764617821553447878227881, 1.97799695347965926190689532453, 2.57750537949978826207225020711, 2.90431789479753584336146968192, 3.46198825228872491364336373573, 3.58733371813895854385652933655, 4.16643633425528737547236867545, 4.27055051262382691251328816138, 4.87661083685185536396222788999, 4.93463182735863957646733011930, 5.52296296149824219893407750670, 5.79160217109100923462112886668, 6.02262762301723452008359783956, 6.44570425849597626422251060666, 6.80666350852720939488415435772, 7.01428659592894796935823290869, 7.57029916872560515994074720650, 7.63262370111043843131704370241
