| L(s) = 1 | − 8·2-s + 48·4-s + 108·5-s − 256·8-s − 864·10-s + 124·11-s − 720·13-s + 1.28e3·16-s + 612·17-s − 2.08e3·19-s + 5.18e3·20-s − 992·22-s − 772·23-s + 2.54e3·25-s + 5.76e3·26-s + 4.59e3·29-s − 9.79e3·31-s − 6.14e3·32-s − 4.89e3·34-s − 5.99e3·37-s + 1.67e4·38-s − 2.76e4·40-s + 2.01e4·41-s − 1.13e3·43-s + 5.95e3·44-s + 6.17e3·46-s + 3.69e4·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.93·5-s − 1.41·8-s − 2.73·10-s + 0.308·11-s − 1.18·13-s + 5/4·16-s + 0.513·17-s − 1.32·19-s + 2.89·20-s − 0.436·22-s − 0.304·23-s + 0.815·25-s + 1.67·26-s + 1.01·29-s − 1.83·31-s − 1.06·32-s − 0.726·34-s − 0.719·37-s + 1.87·38-s − 2.73·40-s + 1.87·41-s − 0.0936·43-s + 0.463·44-s + 0.430·46-s + 2.43·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.770909927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.770909927\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 108 T + 9116 T^{2} - 108 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 124 T + 325298 T^{2} - 124 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 720 T + 716504 T^{2} + 720 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 36 p T - 508 p^{2} T^{2} - 36 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2088 T + 3803406 T^{2} + 2088 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 772 T + 11336234 T^{2} + 772 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4592 T + 40196882 T^{2} - 4592 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9792 T + 81095990 T^{2} + 9792 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5992 T + 51114522 T^{2} + 5992 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 20196 T + 324853604 T^{2} - 20196 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1136 T + 157222710 T^{2} + 1136 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 36936 T + 775004390 T^{2} - 36936 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16708 T + 904428110 T^{2} - 16708 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 74592 T + 2772859406 T^{2} - 74592 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18648 T + 1674996936 T^{2} - 18648 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 67344 T + 3672053798 T^{2} - 67344 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 76548 T + 3345343306 T^{2} + 76548 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 648 p T + 4525629840 T^{2} + 648 p^{6} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 140656 T + 10301739582 T^{2} - 140656 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 94104 T + 8432170262 T^{2} - 94104 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17604 T + 7609766564 T^{2} + 17604 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 85176 T + 11802571296 T^{2} + 85176 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.472503188012700066810360604200, −9.341636006042151708930037484186, −8.782324333235980265062935343886, −8.692436693148307000586525485871, −7.74566552427976090280715307347, −7.74493336115996060968428493093, −6.94309671190992890840851870470, −6.82215709381076973917347763282, −6.16581956887245286406680203949, −5.84886188322078309867486904509, −5.40955901082344166980357755073, −5.09278990377611430610572666719, −4.02834082352546749636615660712, −3.84331071755162911938538952198, −2.69242444785111511072675101493, −2.42530479677957377260222957423, −2.01421025767390383983555830143, −1.69724898783489321214959049244, −0.77705486440398958767738198760, −0.52683638553798236582672461950,
0.52683638553798236582672461950, 0.77705486440398958767738198760, 1.69724898783489321214959049244, 2.01421025767390383983555830143, 2.42530479677957377260222957423, 2.69242444785111511072675101493, 3.84331071755162911938538952198, 4.02834082352546749636615660712, 5.09278990377611430610572666719, 5.40955901082344166980357755073, 5.84886188322078309867486904509, 6.16581956887245286406680203949, 6.82215709381076973917347763282, 6.94309671190992890840851870470, 7.74493336115996060968428493093, 7.74566552427976090280715307347, 8.692436693148307000586525485871, 8.782324333235980265062935343886, 9.341636006042151708930037484186, 9.472503188012700066810360604200
