| L(s) = 1 | − 6·3-s − 4·5-s + 4·7-s + 27·9-s + 28·11-s − 26·13-s + 24·15-s − 36·17-s + 44·19-s − 24·21-s − 8·23-s − 226·25-s − 108·27-s − 204·29-s − 164·31-s − 168·33-s − 16·35-s − 668·37-s + 156·39-s − 100·41-s − 272·43-s − 108·45-s + 60·47-s − 566·49-s + 216·51-s − 708·53-s − 112·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.357·5-s + 0.215·7-s + 9-s + 0.767·11-s − 0.554·13-s + 0.413·15-s − 0.513·17-s + 0.531·19-s − 0.249·21-s − 0.0725·23-s − 1.80·25-s − 0.769·27-s − 1.30·29-s − 0.950·31-s − 0.886·33-s − 0.0772·35-s − 2.96·37-s + 0.640·39-s − 0.380·41-s − 0.964·43-s − 0.357·45-s + 0.186·47-s − 1.65·49-s + 0.593·51-s − 1.83·53-s − 0.274·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 242 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 582 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 28 T + 2426 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 36 T + 2038 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 44 T - 498 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 8798 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 204 T + 52270 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 164 T + 66294 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 668 T + 212814 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 100 T - 2230 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 272 T + 137142 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 60 T + 203746 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 708 T + 312478 T^{2} + 708 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 180 T + 395626 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1068 T + 726830 T^{2} + 1068 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 420 T + 207854 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 436 T + 749474 T^{2} - 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 412 T + 163398 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 672 T + 559646 T^{2} - 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 124 T + 501530 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 140 T + 1088138 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 188 T - 343530 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.90042318070465831834560823108, −10.88993503049656563820922312661, −10.04122213100449052792144620156, −9.733809704043121255205222347913, −9.176314488915552031245581611931, −8.798266037766021627463178879892, −7.81676760546915516721479235184, −7.75183967280174308472944621727, −6.95895285607915455130138091454, −6.71557368891686097648852347573, −5.91572732978783910958676792490, −5.64847200944659758610050754833, −4.76482157037134973651968638957, −4.68599079511752888023976641419, −3.52779358656910079028742869930, −3.49141092879221520172440692377, −1.75300276092692768523352777055, −1.72098342268013594403718883638, 0, 0,
1.72098342268013594403718883638, 1.75300276092692768523352777055, 3.49141092879221520172440692377, 3.52779358656910079028742869930, 4.68599079511752888023976641419, 4.76482157037134973651968638957, 5.64847200944659758610050754833, 5.91572732978783910958676792490, 6.71557368891686097648852347573, 6.95895285607915455130138091454, 7.75183967280174308472944621727, 7.81676760546915516721479235184, 8.798266037766021627463178879892, 9.176314488915552031245581611931, 9.733809704043121255205222347913, 10.04122213100449052792144620156, 10.88993503049656563820922312661, 10.90042318070465831834560823108
