| L(s) = 1 | + 2·9-s + 24·19-s − 30·25-s − 136·31-s + 68·49-s + 296·61-s + 312·79-s − 77·81-s + 296·109-s + 324·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 388·169-s + 48·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2/9·9-s + 1.26·19-s − 6/5·25-s − 4.38·31-s + 1.38·49-s + 4.85·61-s + 3.94·79-s − 0.950·81-s + 2.71·109-s + 2.67·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.29·169-s + 0.280·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.581698761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.581698761\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 p T^{2} + p^{4} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 402 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 1038 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3042 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2914 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4398 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3998 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2718 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 514 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7202 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 7522 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 3198 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15522 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17794 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.474650074506237418024967678010, −8.373462495102438507997040234245, −8.304753875661157573872469685996, −7.59472639468324016279764599133, −7.48461551436711410171882239417, −7.37274004142461945432821216038, −7.05456009327492970399079313917, −6.97291429084686988208839233027, −6.44437927171065323148127677280, −6.14750962579094369219611751526, −5.70908979432120547810510263524, −5.63065868382655233141232291160, −5.27800358351991013344095605343, −5.20755201203946116914070106762, −4.72115806027518190795961766970, −4.29089109209537392358797095286, −3.80620736054263723308705339874, −3.68139811188839564369933535272, −3.52913037406747222091197142703, −3.07556930441703235749610654358, −2.31988866468879359275629985219, −1.96939373374462807014006082662, −1.95615244816119335513256575703, −0.966241623072995517799440071720, −0.49902437007466475314303390076,
0.49902437007466475314303390076, 0.966241623072995517799440071720, 1.95615244816119335513256575703, 1.96939373374462807014006082662, 2.31988866468879359275629985219, 3.07556930441703235749610654358, 3.52913037406747222091197142703, 3.68139811188839564369933535272, 3.80620736054263723308705339874, 4.29089109209537392358797095286, 4.72115806027518190795961766970, 5.20755201203946116914070106762, 5.27800358351991013344095605343, 5.63065868382655233141232291160, 5.70908979432120547810510263524, 6.14750962579094369219611751526, 6.44437927171065323148127677280, 6.97291429084686988208839233027, 7.05456009327492970399079313917, 7.37274004142461945432821216038, 7.48461551436711410171882239417, 7.59472639468324016279764599133, 8.304753875661157573872469685996, 8.373462495102438507997040234245, 8.474650074506237418024967678010
