| L(s) = 1 | + 5·3-s + 15·5-s − 29·7-s + 27·9-s + 51·11-s − 53·13-s + 75·15-s − 51·17-s + 145·19-s − 145·21-s + 216·23-s + 125·25-s + 280·27-s − 396·29-s − 124·31-s + 255·33-s − 435·35-s + 55·37-s − 265·39-s − 39·41-s + 217·43-s + 405·45-s + 1.00e3·47-s + 343·49-s − 255·51-s − 237·53-s + 765·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s + 1.34·5-s − 1.56·7-s + 9-s + 1.39·11-s − 1.13·13-s + 1.29·15-s − 0.727·17-s + 1.75·19-s − 1.50·21-s + 1.95·23-s + 25-s + 1.99·27-s − 2.53·29-s − 0.718·31-s + 1.34·33-s − 2.10·35-s + 0.244·37-s − 1.08·39-s − 0.148·41-s + 0.769·43-s + 1.34·45-s + 3.12·47-s + 49-s − 0.700·51-s − 0.614·53-s + 1.87·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.684377606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.684377606\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 31 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T - 2 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 p T + 4 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 29 T + 498 T^{2} + 29 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 51 T + 1270 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 53 T + 612 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 p T - 8 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 145 T + 14166 T^{2} - 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 55 T - 47628 T^{2} - 55 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 39 T - 67400 T^{2} + 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 217 T - 32418 T^{2} - 217 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 504 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 237 T - 92708 T^{2} + 237 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 597 T + 151030 T^{2} - 597 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 466 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 749 T + 260238 T^{2} + 749 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 321 T - 254870 T^{2} - 321 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 793 T + 239832 T^{2} - 793 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 1217 T + 988050 T^{2} + 1217 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 555 T - 263762 T^{2} - 555 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 378 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1070 T + p^{3} T^{2} )^{2} \) |
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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
−13.07801116080491758500047915576, −12.91925158582011081544174509236, −12.42509503778786845359081924434, −11.78178699995287012246224750745, −10.97287805649269812241432841812, −10.42007009906316648039681392183, −9.767280083210063945462366233794, −9.307572713288341011608608350999, −9.167954829501575427924128677101, −8.946467009436853455538719705570, −7.37952036965690991038176824485, −7.29284962641833792884670510295, −6.72347216269254547419069962059, −5.96745485608797165473502764129, −5.37361308200593026674738909726, −4.44695695231254433285938555412, −3.51017479665169200658415147616, −2.94564943221072461232430167602, −2.13232973208578404616792345410, −1.02691126080557645004569121374,
1.02691126080557645004569121374, 2.13232973208578404616792345410, 2.94564943221072461232430167602, 3.51017479665169200658415147616, 4.44695695231254433285938555412, 5.37361308200593026674738909726, 5.96745485608797165473502764129, 6.72347216269254547419069962059, 7.29284962641833792884670510295, 7.37952036965690991038176824485, 8.946467009436853455538719705570, 9.167954829501575427924128677101, 9.307572713288341011608608350999, 9.767280083210063945462366233794, 10.42007009906316648039681392183, 10.97287805649269812241432841812, 11.78178699995287012246224750745, 12.42509503778786845359081924434, 12.91925158582011081544174509236, 13.07801116080491758500047915576
