| L(s) = 1 | + 6·3-s − 52·7-s − 45·9-s + 100·13-s + 716·19-s − 312·21-s + 962·25-s − 756·27-s + 1.48e3·31-s + 3.74e3·37-s + 600·39-s + 524·43-s − 2.77e3·49-s + 4.29e3·57-s − 2.97e3·61-s + 2.34e3·63-s + 8.97e3·67-s + 580·73-s + 5.77e3·75-s − 1.96e4·79-s − 891·81-s − 5.20e3·91-s + 8.90e3·93-s − 956·97-s − 4.27e3·103-s − 9.50e3·109-s + 2.24e4·111-s + ⋯ |
| L(s) = 1 | + 2/3·3-s − 1.06·7-s − 5/9·9-s + 0.591·13-s + 1.98·19-s − 0.707·21-s + 1.53·25-s − 1.03·27-s + 1.54·31-s + 2.73·37-s + 0.394·39-s + 0.283·43-s − 1.15·49-s + 1.32·57-s − 0.798·61-s + 0.589·63-s + 1.99·67-s + 0.108·73-s + 1.02·75-s − 3.14·79-s − 0.135·81-s − 0.627·91-s + 1.02·93-s − 0.101·97-s − 0.403·103-s − 0.799·109-s + 1.82·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.137206147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137206147\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 - 2 p T + p^{4} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 962 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 26 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 15170 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 50 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 125570 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 358 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 420290 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 666238 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 742 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1874 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 155710 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 262 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6879362 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15571010 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 20937410 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 1486 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4486 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 38122562 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 290 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 9818 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 44355074 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 64012610 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 478 T + p^{4} T^{2} )^{2} \) |
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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
−15.20204965682210743390072384405, −14.40178591051526319057073211519, −14.21187085822612066353874435569, −13.38859166727235012750502358465, −13.09334046393543551315473444208, −12.42507136575696730277740945049, −11.53533082944823286741175199295, −11.28335946274877857776375671491, −10.29209476282420477926701228817, −9.506088632646515188568734889633, −9.391435470464792430141062130219, −8.387387165107447761824692180553, −7.921997317401875765604710521739, −6.99571619135460566785308552033, −6.26407226665218984366146259678, −5.53638632688435916432706242113, −4.41391969488967052330068711725, −3.17433542726073925731858863250, −2.84656439844547018661678233560, −0.931900972796667056355333710549,
0.931900972796667056355333710549, 2.84656439844547018661678233560, 3.17433542726073925731858863250, 4.41391969488967052330068711725, 5.53638632688435916432706242113, 6.26407226665218984366146259678, 6.99571619135460566785308552033, 7.921997317401875765604710521739, 8.387387165107447761824692180553, 9.391435470464792430141062130219, 9.506088632646515188568734889633, 10.29209476282420477926701228817, 11.28335946274877857776375671491, 11.53533082944823286741175199295, 12.42507136575696730277740945049, 13.09334046393543551315473444208, 13.38859166727235012750502358465, 14.21187085822612066353874435569, 14.40178591051526319057073211519, 15.20204965682210743390072384405
