L(s) = 1 | + 3·2-s + 3·3-s + 8·4-s + 18·5-s + 9·6-s + 45·8-s + 54·10-s + 36·11-s + 24·12-s − 68·13-s + 54·15-s + 135·16-s − 42·17-s + 124·19-s + 144·20-s + 108·22-s + 135·24-s + 125·25-s − 204·26-s − 27·27-s + 204·29-s + 162·30-s + 160·31-s + 360·32-s + 108·33-s − 126·34-s − 398·37-s + ⋯ |
L(s) = 1 | + 1.06·2-s + 0.577·3-s + 4-s + 1.60·5-s + 0.612·6-s + 1.98·8-s + 1.70·10-s + 0.986·11-s + 0.577·12-s − 1.45·13-s + 0.929·15-s + 2.10·16-s − 0.599·17-s + 1.49·19-s + 1.60·20-s + 1.04·22-s + 1.14·24-s + 25-s − 1.53·26-s − 0.192·27-s + 1.30·29-s + 0.985·30-s + 0.926·31-s + 1.98·32-s + 0.569·33-s − 0.635·34-s − 1.76·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(8.504416037\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.504416037\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 18 T + 199 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 36 T - 35 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 42 T - 3149 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 124 T + 8517 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 102 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 160 T - 4191 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 398 T + 107751 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 318 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 268 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 240 T - 46223 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 498 T + 99127 T^{2} - 498 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 132 T - 187955 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 398 T - 68577 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 92 T - 292299 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 720 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 502 T - 137013 T^{2} - 502 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1024 T + 555537 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 204 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 354 T - 579653 T^{2} + 354 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 286 T + p^{3} T^{2} )^{2} \) |
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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
−13.30624328202591267409369612700, −12.29284372985196174583039302105, −11.84856313581763872790078226621, −11.74873248680591459369783079130, −10.64483611414046764699936459774, −10.25129650512381807787937548984, −9.830715025281998420737590511703, −9.537668631563427251780136069498, −8.570232827982493879575114273638, −8.196729264748521278575518083002, −7.07073344101855824087974020862, −7.03892974771903186343672133942, −6.37777405805474349100844875411, −5.51295160454589611163370816242, −4.98385409893098249065032521892, −4.62594930458511973997169298755, −3.51970122368449259145734785705, −2.85178390003630857089305525650, −1.93437003949380149975697212137, −1.46416920708817542687523475314,
1.46416920708817542687523475314, 1.93437003949380149975697212137, 2.85178390003630857089305525650, 3.51970122368449259145734785705, 4.62594930458511973997169298755, 4.98385409893098249065032521892, 5.51295160454589611163370816242, 6.37777405805474349100844875411, 7.03892974771903186343672133942, 7.07073344101855824087974020862, 8.196729264748521278575518083002, 8.570232827982493879575114273638, 9.537668631563427251780136069498, 9.830715025281998420737590511703, 10.25129650512381807787937548984, 10.64483611414046764699936459774, 11.74873248680591459369783079130, 11.84856313581763872790078226621, 12.29284372985196174583039302105, 13.30624328202591267409369612700