| L(s) = 1 | + 2·3-s − 4·5-s + 3·9-s − 2·11-s − 2·13-s − 8·15-s − 4·17-s − 4·23-s + 4·25-s + 4·27-s − 4·29-s − 8·31-s − 4·33-s − 4·37-s − 4·39-s − 4·41-s + 8·43-s − 12·45-s − 16·47-s − 6·49-s − 8·51-s − 4·53-s + 8·55-s − 12·61-s + 8·65-s − 8·69-s − 16·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 9-s − 0.603·11-s − 0.554·13-s − 2.06·15-s − 0.970·17-s − 0.834·23-s + 4/5·25-s + 0.769·27-s − 0.742·29-s − 1.43·31-s − 0.696·33-s − 0.657·37-s − 0.640·39-s − 0.624·41-s + 1.21·43-s − 1.78·45-s − 2.33·47-s − 6/7·49-s − 1.12·51-s − 0.549·53-s + 1.07·55-s − 1.53·61-s + 0.992·65-s − 0.963·69-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2944656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2944656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 76 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 84 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 108 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 286 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−9.072973926620682511713090167805, −8.558741438125610655028376149516, −8.261705529710183695936795015857, −8.049641145042043086743187620029, −7.47940167726704741707714199190, −7.47234145826826407411344162030, −6.90228054035137821589216842073, −6.60577758509284272645584940492, −5.75101791819044595225527520120, −5.52036964081294199530870182021, −4.71468428208679951375031585246, −4.38874398868782403464228891623, −4.12961629161061196741422169654, −3.59522428622544930166536907514, −3.05773656710589124985044704637, −2.88327241892100211030826667504, −1.80198998077609674783874483090, −1.77445035581619133065337275923, 0, 0,
1.77445035581619133065337275923, 1.80198998077609674783874483090, 2.88327241892100211030826667504, 3.05773656710589124985044704637, 3.59522428622544930166536907514, 4.12961629161061196741422169654, 4.38874398868782403464228891623, 4.71468428208679951375031585246, 5.52036964081294199530870182021, 5.75101791819044595225527520120, 6.60577758509284272645584940492, 6.90228054035137821589216842073, 7.47234145826826407411344162030, 7.47940167726704741707714199190, 8.049641145042043086743187620029, 8.261705529710183695936795015857, 8.558741438125610655028376149516, 9.072973926620682511713090167805
