| L(s) = 1 | + 7·4-s + 29·9-s − 108·11-s − 15·16-s − 38·19-s + 498·29-s + 112·31-s + 203·36-s + 480·41-s − 756·44-s + 565·49-s − 390·59-s − 716·61-s − 553·64-s − 492·71-s − 266·76-s + 68·79-s + 112·81-s + 336·89-s − 3.13e3·99-s − 1.45e3·101-s − 778·109-s + 3.48e3·116-s + 6.08e3·121-s + 784·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/8·4-s + 1.07·9-s − 2.96·11-s − 0.234·16-s − 0.458·19-s + 3.18·29-s + 0.648·31-s + 0.939·36-s + 1.82·41-s − 2.59·44-s + 1.64·49-s − 0.860·59-s − 1.50·61-s − 1.08·64-s − 0.822·71-s − 0.401·76-s + 0.0968·79-s + 0.153·81-s + 0.400·89-s − 3.17·99-s − 1.43·101-s − 0.683·109-s + 2.79·116-s + 4.57·121-s + 0.567·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.456837630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.456837630\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 29 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 565 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4273 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1177 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9155 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 249 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38806 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 240 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 120598 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 179422 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 108529 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 195 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 358 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 321995 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 246 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 653425 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1088818 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 168 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1250782 T^{2} + p^{6} T^{4} \) |
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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
−10.80314763580801891553656838867, −10.51117018305138200126054120160, −10.08420992789370395966068649220, −9.733329450671153344174543374335, −8.896512592007326063975816046177, −8.548908294762682033419836370368, −7.81864591517462402376587139218, −7.71014582710734509165631205051, −7.31444853595564664213861105993, −6.51055391557236485867852902912, −6.41161682205193025384586499593, −5.60910596357799544439445874679, −5.13943368524502442667515847757, −4.39976894194002726261044658211, −4.39061502826092428991832276329, −3.01816408059376240564816872620, −2.65800746886934216575040268246, −2.38749732175947229116550897311, −1.37157049375896041740276200555, −0.49349360468485387415793609574,
0.49349360468485387415793609574, 1.37157049375896041740276200555, 2.38749732175947229116550897311, 2.65800746886934216575040268246, 3.01816408059376240564816872620, 4.39061502826092428991832276329, 4.39976894194002726261044658211, 5.13943368524502442667515847757, 5.60910596357799544439445874679, 6.41161682205193025384586499593, 6.51055391557236485867852902912, 7.31444853595564664213861105993, 7.71014582710734509165631205051, 7.81864591517462402376587139218, 8.548908294762682033419836370368, 8.896512592007326063975816046177, 9.733329450671153344174543374335, 10.08420992789370395966068649220, 10.51117018305138200126054120160, 10.80314763580801891553656838867
