| L(s) = 1 | + 3-s − 4·5-s − 5·7-s + 9·9-s − 21·11-s − 26·13-s − 4·15-s + 25·17-s − 21·19-s − 5·21-s − 15·23-s − 38·25-s + 26·27-s − 39·29-s − 20·31-s − 21·33-s + 20·35-s − 61·37-s − 26·39-s + 3·41-s + 41·43-s − 36·45-s − 164·47-s + 49·49-s + 25·51-s + 84·55-s − 21·57-s + ⋯ |
| L(s) = 1 | + 1/3·3-s − 4/5·5-s − 5/7·7-s + 9-s − 1.90·11-s − 2·13-s − 0.266·15-s + 1.47·17-s − 1.10·19-s − 0.238·21-s − 0.652·23-s − 1.51·25-s + 0.962·27-s − 1.34·29-s − 0.645·31-s − 0.636·33-s + 4/7·35-s − 1.64·37-s − 2/3·39-s + 3/41·41-s + 0.953·43-s − 4/5·45-s − 3.48·47-s + 49-s + 0.490·51-s + 1.52·55-s − 0.368·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03151317709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03151317709\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 8 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 5 T - 24 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 21 T + 268 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T + 336 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 21 T + 508 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 15 T + 604 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 39 T + 1348 T^{2} + 39 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 61 T + 2352 T^{2} + 61 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T + 1684 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 41 T - 168 T^{2} - 41 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3890 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 93 T + 6364 T^{2} + 93 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 105 T + 7396 T^{2} - 105 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 21 T + 4636 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 29 T - 4200 T^{2} + 29 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 8930 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 3070 T^{2} + p^{4} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 69 T + 9508 T^{2} + 69 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 3 T + 9412 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.29040295736550283285667293205, −10.44135609810443481103088926613, −10.33343532601651005766173931593, −9.885735228627100200078535242408, −9.640426674169075849771594702961, −9.039102680267200962145239292550, −8.183338022836763910977438536758, −7.906812469792846892192862485741, −7.68297991626566231873232377160, −7.08598502259984240722640107972, −6.85992885243061578066788485321, −5.68731308123808046635422054628, −5.64998744928129489452084170215, −4.73252169514818720679078227558, −4.48109302048125822098895603557, −3.49201665856261041914624576306, −3.32560028170798888975323935755, −2.32577728057659925821661903861, −1.88035305647597642884520458780, −0.06938897454619178712173752298,
0.06938897454619178712173752298, 1.88035305647597642884520458780, 2.32577728057659925821661903861, 3.32560028170798888975323935755, 3.49201665856261041914624576306, 4.48109302048125822098895603557, 4.73252169514818720679078227558, 5.64998744928129489452084170215, 5.68731308123808046635422054628, 6.85992885243061578066788485321, 7.08598502259984240722640107972, 7.68297991626566231873232377160, 7.906812469792846892192862485741, 8.183338022836763910977438536758, 9.039102680267200962145239292550, 9.640426674169075849771594702961, 9.885735228627100200078535242408, 10.33343532601651005766173931593, 10.44135609810443481103088926613, 11.29040295736550283285667293205
