| L(s) = 1 | + 18·5-s − 8·7-s − 36·11-s + 10·13-s + 36·17-s − 200·19-s − 72·23-s + 125·25-s + 234·29-s + 16·31-s − 144·35-s − 452·37-s − 90·41-s − 452·43-s − 432·47-s + 343·49-s + 828·53-s − 648·55-s + 684·59-s − 422·61-s + 180·65-s − 332·67-s − 720·71-s + 52·73-s + 288·77-s − 512·79-s + 1.18e3·83-s + ⋯ |
| L(s) = 1 | + 1.60·5-s − 0.431·7-s − 0.986·11-s + 0.213·13-s + 0.513·17-s − 2.41·19-s − 0.652·23-s + 25-s + 1.49·29-s + 0.0926·31-s − 0.695·35-s − 2.00·37-s − 0.342·41-s − 1.60·43-s − 1.34·47-s + 49-s + 2.14·53-s − 1.58·55-s + 1.50·59-s − 0.885·61-s + 0.343·65-s − 0.605·67-s − 1.20·71-s + 0.0833·73-s + 0.426·77-s − 0.729·79-s + 1.57·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.069756140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.069756140\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 18 T + 199 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T - 279 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 36 T - 35 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 10 T - 2097 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 72 T - 6983 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 234 T + 30367 T^{2} - 234 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 16 T - 29535 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 226 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 90 T - 60821 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 452 T + 124797 T^{2} + 452 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 432 T + 82801 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 414 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 684 T + 262477 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 422 T - 48897 T^{2} + 422 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 332 T - 190539 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 512 T - 230895 T^{2} + 512 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1188 T + 839557 T^{2} - 1188 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1054 T + 198243 T^{2} - 1054 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.43848592923593987938470684567, −10.61216802299883852630101740874, −10.29553006012149508293066500312, −10.08654322941997610793102016517, −9.943750440043298503876816067932, −8.809995159413653243515033930688, −8.734837888530627479987342509794, −8.366175446852473700205705169528, −7.57756382248117425255313414612, −6.83679398541803652277218034304, −6.57020862355900790720401569338, −5.93504526661038324901158084350, −5.67128313361226790105324766739, −4.95928872788410146533922219163, −4.46336580384367985116580119687, −3.56358895733880241361126288584, −2.90259414787113244866039686789, −2.05870349678110662650084651497, −1.84633148894208864292107148723, −0.49058899041754162739998370185,
0.49058899041754162739998370185, 1.84633148894208864292107148723, 2.05870349678110662650084651497, 2.90259414787113244866039686789, 3.56358895733880241361126288584, 4.46336580384367985116580119687, 4.95928872788410146533922219163, 5.67128313361226790105324766739, 5.93504526661038324901158084350, 6.57020862355900790720401569338, 6.83679398541803652277218034304, 7.57756382248117425255313414612, 8.366175446852473700205705169528, 8.734837888530627479987342509794, 8.809995159413653243515033930688, 9.943750440043298503876816067932, 10.08654322941997610793102016517, 10.29553006012149508293066500312, 10.61216802299883852630101740874, 11.43848592923593987938470684567
