| L(s) = 1 | + 7.06e3·7-s − 8.36e4·13-s + 7.26e4·19-s + 7.26e5·25-s + 9.42e5·31-s − 6.01e6·37-s − 7.24e6·43-s + 2.58e7·49-s − 1.08e7·61-s + 1.22e7·67-s − 9.80e7·73-s − 1.67e7·79-s − 5.90e8·91-s + 4.08e7·97-s + 5.96e7·103-s − 9.77e7·109-s + 2.15e7·121-s + 127-s + 131-s + 5.12e8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2.94·7-s − 2.92·13-s + 0.557·19-s + 1.86·25-s + 1.02·31-s − 3.20·37-s − 2.11·43-s + 4.49·49-s − 0.785·61-s + 0.607·67-s − 3.45·73-s − 0.429·79-s − 8.61·91-s + 0.461·97-s + 0.529·103-s − 0.692·109-s + 0.100·121-s + 1.63·133-s + 4.43·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.601791507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.601791507\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 29072 p^{2} T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3532 T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 21566114 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 41824 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4967349824 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 36304 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14462450590 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 928028865104 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 471196 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3007402 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 13027466643584 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 3623720 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 11446895562722 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 18909552109520 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 286434976404290 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 5440630 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 6121576 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 842318136694370 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 49031152 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8357756 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1863467106641954 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3648102662700160 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 20431328 T + p^{8} T^{2} )^{2} \) |
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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
−12.05985090599246976207946421410, −11.52719453318749662175185872020, −10.77206183790011807924976753334, −10.21968839638803288724706074204, −10.09623114594643711308210121444, −9.007004090213987078392465153978, −8.701549510722226829083141163075, −8.089176051272657752388306856272, −7.66057695202965878436216998668, −7.13041882802259757401924908723, −6.79295182299146154192987433184, −5.39167679702558825098620069790, −5.09669818759826932078115905796, −4.81357497707326662509348053811, −4.40135134084475484292679864346, −3.12877426045037453538169189209, −2.52247906336083337023075393012, −1.66090808520873578864692216797, −1.51312258171597204707617362831, −0.39430406727137785130052328246,
0.39430406727137785130052328246, 1.51312258171597204707617362831, 1.66090808520873578864692216797, 2.52247906336083337023075393012, 3.12877426045037453538169189209, 4.40135134084475484292679864346, 4.81357497707326662509348053811, 5.09669818759826932078115905796, 5.39167679702558825098620069790, 6.79295182299146154192987433184, 7.13041882802259757401924908723, 7.66057695202965878436216998668, 8.089176051272657752388306856272, 8.701549510722226829083141163075, 9.007004090213987078392465153978, 10.09623114594643711308210121444, 10.21968839638803288724706074204, 10.77206183790011807924976753334, 11.52719453318749662175185872020, 12.05985090599246976207946421410
