L(s) = 1 | − 2·3-s − 4·5-s + 3·9-s − 4·11-s + 8·15-s + 12·17-s + 8·19-s + 4·23-s + 4·25-s − 4·27-s + 8·33-s − 8·37-s + 12·41-s − 12·45-s − 8·47-s − 24·51-s + 4·53-s + 16·55-s − 16·57-s − 8·61-s − 8·69-s + 4·71-s + 24·73-s − 8·75-s − 16·79-s + 5·81-s − 8·83-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.78·5-s + 9-s − 1.20·11-s + 2.06·15-s + 2.91·17-s + 1.83·19-s + 0.834·23-s + 4/5·25-s − 0.769·27-s + 1.39·33-s − 1.31·37-s + 1.87·41-s − 1.78·45-s − 1.16·47-s − 3.36·51-s + 0.549·53-s + 2.15·55-s − 2.11·57-s − 1.02·61-s − 0.963·69-s + 0.474·71-s + 2.80·73-s − 0.923·75-s − 1.80·79-s + 5/9·81-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.186335943\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.186335943\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 12 T + 4 p T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 24 T + 272 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 192 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.74008040769371072735520884509, −7.71212231384432633819634952408, −7.16784655540374758434895804035, −7.14541147584709354759666860321, −6.38589310716060449822510360685, −6.27585159398255304030811152763, −5.52150892788813761471739189031, −5.42975117689147303756168167434, −5.16632580600902070163892745281, −5.05691908711372113357746931652, −4.34561839721266064699510588704, −4.03156196042972912551796396345, −3.57833832579480186289969532436, −3.39316266636491872321900653410, −2.94591525656567255632870769662, −2.64952383594680782439201515766, −1.71216630829692731389855337626, −1.25002761239093468326579984739, −0.76600126451800262411021570334, −0.40811534615367071718572485882,
0.40811534615367071718572485882, 0.76600126451800262411021570334, 1.25002761239093468326579984739, 1.71216630829692731389855337626, 2.64952383594680782439201515766, 2.94591525656567255632870769662, 3.39316266636491872321900653410, 3.57833832579480186289969532436, 4.03156196042972912551796396345, 4.34561839721266064699510588704, 5.05691908711372113357746931652, 5.16632580600902070163892745281, 5.42975117689147303756168167434, 5.52150892788813761471739189031, 6.27585159398255304030811152763, 6.38589310716060449822510360685, 7.14541147584709354759666860321, 7.16784655540374758434895804035, 7.71212231384432633819634952408, 7.74008040769371072735520884509