| L(s) = 1 | − 3·4-s + 18·7-s − 68·13-s − 55·16-s − 202·19-s − 54·28-s − 6·31-s + 134·37-s + 274·43-s − 443·49-s + 204·52-s − 1.12e3·61-s + 357·64-s − 2.08e3·67-s + 1.00e3·73-s + 606·76-s + 1.23e3·79-s − 1.22e3·91-s + 38·97-s − 298·103-s − 2.35e3·109-s − 990·112-s − 2.45e3·121-s + 18·124-s + 127-s + 131-s − 3.63e3·133-s + ⋯ |
| L(s) = 1 | − 3/8·4-s + 0.971·7-s − 1.45·13-s − 0.859·16-s − 2.43·19-s − 0.364·28-s − 0.0347·31-s + 0.595·37-s + 0.971·43-s − 1.29·49-s + 0.544·52-s − 2.36·61-s + 0.697·64-s − 3.80·67-s + 1.61·73-s + 0.914·76-s + 1.75·79-s − 1.41·91-s + 0.0397·97-s − 0.285·103-s − 2.06·109-s − 0.835·112-s − 1.84·121-s + 0.0130·124-s + 0.000698·127-s + 0.000666·131-s − 2.37·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 9 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2454 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 9774 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 101 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 12634 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 21270 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 67 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 97074 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 137 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 194334 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 161718 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 350574 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 563 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 1044 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 98010 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 503 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 615 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1086946 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 462238 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p^{3} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.797831410781011487116710953923, −9.378846801564981211213098478464, −9.023143744690717401032053349877, −8.694442914933897526915051853839, −7.975935884957555559527193911512, −7.86439059481285467943817051470, −7.37993672431339052953696958473, −6.68500986746100108533230295730, −6.37641982864820342821759461366, −5.89392193243502019414611166455, −5.04310895801708967401223201553, −4.85017976376889857869666555976, −4.31854178512343879534194241926, −4.10110347115195155218451467304, −3.07075331426686902174164930167, −2.43137051438313020062994587431, −2.02674962359235680984445124316, −1.31521111768489924789725266447, 0, 0,
1.31521111768489924789725266447, 2.02674962359235680984445124316, 2.43137051438313020062994587431, 3.07075331426686902174164930167, 4.10110347115195155218451467304, 4.31854178512343879534194241926, 4.85017976376889857869666555976, 5.04310895801708967401223201553, 5.89392193243502019414611166455, 6.37641982864820342821759461366, 6.68500986746100108533230295730, 7.37993672431339052953696958473, 7.86439059481285467943817051470, 7.975935884957555559527193911512, 8.694442914933897526915051853839, 9.023143744690717401032053349877, 9.378846801564981211213098478464, 9.797831410781011487116710953923
