| L(s) = 1 | − 16·5-s − 14·7-s + 4·11-s + 52·13-s − 56·17-s − 32·19-s − 44·23-s + 114·25-s − 20·29-s + 296·31-s + 224·35-s + 172·37-s − 224·41-s + 144·43-s − 344·47-s + 147·49-s − 364·53-s − 64·55-s + 464·59-s + 620·61-s − 832·65-s − 904·67-s + 508·71-s + 12·73-s − 56·77-s − 344·79-s + 280·83-s + ⋯ |
| L(s) = 1 | − 1.43·5-s − 0.755·7-s + 0.109·11-s + 1.10·13-s − 0.798·17-s − 0.386·19-s − 0.398·23-s + 0.911·25-s − 0.128·29-s + 1.71·31-s + 1.08·35-s + 0.764·37-s − 0.853·41-s + 0.510·43-s − 1.06·47-s + 3/7·49-s − 0.943·53-s − 0.156·55-s + 1.02·59-s + 1.30·61-s − 1.58·65-s − 1.64·67-s + 0.849·71-s + 0.0192·73-s − 0.0828·77-s − 0.489·79-s + 0.370·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 16 T + 142 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 2494 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 56 T + 9062 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 32 T + 13286 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 44 T + 4006 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 20 T + 48190 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 296 T + 64286 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 172 T + 64670 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 224 T + 129574 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 144 T + 120166 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 344 T + 231038 T^{2} + 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 364 T + 286846 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 464 T + 265750 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 620 T + 494334 T^{2} - 620 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 904 T + 651030 T^{2} + 904 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 508 T + 544870 T^{2} - 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 276518 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 344 T + 1009470 T^{2} + 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 280 T + 458662 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 848 T + 1585414 T^{2} + 848 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1372 T + 2047574 T^{2} - 1372 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.612791285949073706531587505363, −8.336039623543359076497759131564, −7.75260816702862275342486642993, −7.59310936778953590356192079917, −6.97532054193513689238729953243, −6.61469216922947397334077600198, −6.21369287517723267387732270180, −6.12623262067545515497132809898, −5.31445237440039976267991640712, −4.88512386852442788292717745018, −4.27591125271376112939110390963, −4.18016376026045524938180645562, −3.45928835918163862253415788683, −3.45679015279205647687366515238, −2.62835939304851292136039883071, −2.29450896704680708529025402909, −1.34082979172369838052517997275, −0.940307032655739256480470979367, 0, 0,
0.940307032655739256480470979367, 1.34082979172369838052517997275, 2.29450896704680708529025402909, 2.62835939304851292136039883071, 3.45679015279205647687366515238, 3.45928835918163862253415788683, 4.18016376026045524938180645562, 4.27591125271376112939110390963, 4.88512386852442788292717745018, 5.31445237440039976267991640712, 6.12623262067545515497132809898, 6.21369287517723267387732270180, 6.61469216922947397334077600198, 6.97532054193513689238729953243, 7.59310936778953590356192079917, 7.75260816702862275342486642993, 8.336039623543359076497759131564, 8.612791285949073706531587505363
