| L(s) = 1 | + 2·5-s − 6·9-s + 8·11-s + 4·13-s + 4·17-s − 2·19-s + 3·25-s + 4·29-s + 4·37-s + 4·41-s − 12·45-s − 16·47-s − 6·49-s + 20·53-s + 16·55-s − 8·59-s + 12·61-s + 8·65-s + 20·73-s + 27·81-s − 16·83-s + 8·85-s − 12·89-s − 4·95-s + 20·97-s − 48·99-s + 28·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s + 2.41·11-s + 1.10·13-s + 0.970·17-s − 0.458·19-s + 3/5·25-s + 0.742·29-s + 0.657·37-s + 0.624·41-s − 1.78·45-s − 2.33·47-s − 6/7·49-s + 2.74·53-s + 2.15·55-s − 1.04·59-s + 1.53·61-s + 0.992·65-s + 2.34·73-s + 3·81-s − 1.75·83-s + 0.867·85-s − 1.27·89-s − 0.410·95-s + 2.03·97-s − 4.82·99-s + 2.78·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.036327834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.036327834\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 20 T + 198 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_4$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 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
−8.805233075571186055048632497434, −8.660588447112110479282914369416, −8.241020644193179124639666027941, −8.066212402028142134790658834844, −7.29644067442498535136339469534, −6.90682651057626838526529177761, −6.40350774393606988538874210083, −6.26500236092917154340128653497, −5.95623869162944919487791266276, −5.68430432644761039515805182789, −5.04308427901035229952216082834, −4.81795867000703910332542638572, −3.95875585348658232981967543128, −3.83645826523414314899856380878, −3.23695499460751725303895006977, −2.98917693948920840550455376871, −2.24313892026549684212625283889, −1.83626422047082080787737782402, −1.10753617760283147762944055380, −0.73582021518784170435792738917,
0.73582021518784170435792738917, 1.10753617760283147762944055380, 1.83626422047082080787737782402, 2.24313892026549684212625283889, 2.98917693948920840550455376871, 3.23695499460751725303895006977, 3.83645826523414314899856380878, 3.95875585348658232981967543128, 4.81795867000703910332542638572, 5.04308427901035229952216082834, 5.68430432644761039515805182789, 5.95623869162944919487791266276, 6.26500236092917154340128653497, 6.40350774393606988538874210083, 6.90682651057626838526529177761, 7.29644067442498535136339469534, 8.066212402028142134790658834844, 8.241020644193179124639666027941, 8.660588447112110479282914369416, 8.805233075571186055048632497434
