| L(s) = 1 | + 2·3-s − 3·5-s − 7-s − 3·9-s − 2·11-s − 2·13-s − 6·15-s − 4·17-s + 10·19-s − 2·21-s + 2·23-s − 2·25-s − 14·27-s − 2·31-s − 4·33-s + 3·35-s − 4·37-s − 4·39-s + 9·41-s + 17·43-s + 9·45-s + 9·47-s − 2·49-s − 8·51-s + 3·53-s + 6·55-s + 20·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s − 0.377·7-s − 9-s − 0.603·11-s − 0.554·13-s − 1.54·15-s − 0.970·17-s + 2.29·19-s − 0.436·21-s + 0.417·23-s − 2/5·25-s − 2.69·27-s − 0.359·31-s − 0.696·33-s + 0.507·35-s − 0.657·37-s − 0.640·39-s + 1.40·41-s + 2.59·43-s + 1.34·45-s + 1.31·47-s − 2/7·49-s − 1.12·51-s + 0.412·53-s + 0.809·55-s + 2.64·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29767936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29767936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.026455188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026455188\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 17 T + 157 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 83 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 11 T + 141 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 23 T + 297 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.103533204861097328257598756473, −8.037935448136260968277502232757, −7.71319213082133703577320086537, −7.48131108036914528146401562867, −7.01941526928528575148956463115, −6.88773568940097739793241507893, −5.99550482440426058541273965092, −5.84737525669052350705038916276, −5.34992777801140335146249834974, −5.29618658242904767246641131578, −4.41789325081809483506027821191, −4.26986527955692702408312361297, −3.74498432990542155170220756396, −3.44362356067099922142625985377, −2.92576314860851651599061960089, −2.85410022436810456800904781653, −2.24838541744907421234788466621, −1.94345984712355479715586510771, −0.73870672293201901698670848464, −0.48726591562902766335851586498,
0.48726591562902766335851586498, 0.73870672293201901698670848464, 1.94345984712355479715586510771, 2.24838541744907421234788466621, 2.85410022436810456800904781653, 2.92576314860851651599061960089, 3.44362356067099922142625985377, 3.74498432990542155170220756396, 4.26986527955692702408312361297, 4.41789325081809483506027821191, 5.29618658242904767246641131578, 5.34992777801140335146249834974, 5.84737525669052350705038916276, 5.99550482440426058541273965092, 6.88773568940097739793241507893, 7.01941526928528575148956463115, 7.48131108036914528146401562867, 7.71319213082133703577320086537, 8.037935448136260968277502232757, 8.103533204861097328257598756473
