| L(s) = 1 | − 16·4-s − 708·11-s + 256·16-s − 3.59e3·19-s − 1.15e4·29-s + 2.03e4·31-s + 2.59e4·41-s + 1.13e4·44-s + 2.40e4·49-s + 7.90e3·59-s + 1.92e3·61-s − 4.09e3·64-s + 1.12e5·71-s + 5.74e4·76-s + 5.20e4·79-s + 1.46e5·89-s + 3.09e5·101-s − 2.46e5·109-s + 1.84e5·116-s + 5.38e4·121-s − 3.26e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.76·11-s + 1/4·16-s − 2.28·19-s − 2.54·29-s + 3.81·31-s + 2.40·41-s + 0.882·44-s + 10/7·49-s + 0.295·59-s + 0.0662·61-s − 1/8·64-s + 2.64·71-s + 1.14·76-s + 0.939·79-s + 1.96·89-s + 3.01·101-s − 1.98·109-s + 1.27·116-s + 0.334·121-s − 1.90·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.211570071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.211570071\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 p^{4} T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 354 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 579370 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2411998 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1796 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11706286 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5754 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10196 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 107863210 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12960 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 213591862 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 429530014 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 768921190 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3954 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 962 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 p^{4} T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 56148 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3196632914 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 26044 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 858185738 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 73428 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 539356030 T^{2} + p^{10} T^{4} \) |
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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
−10.34219346670654745362660129872, −10.32201929136535336802421754363, −9.442535322036218021477214214738, −9.364117298900427358986604649774, −8.486584387266337560833474699133, −8.353678078345244855121736809398, −7.75318129817252636597864626224, −7.58171582189101263195110343002, −6.69354897610083707626986067754, −6.30745054784354111580924889557, −5.75348664454641273635208613074, −5.33840409146129457554554047888, −4.59940060664626995630977527685, −4.37541871248185439060241844236, −3.75244587907017742849056603563, −2.95222965365990008370619645064, −2.24496994235425324259953094475, −2.14348545924013479149918571766, −0.67309541194531818486015002203, −0.56070629357527069293556852313,
0.56070629357527069293556852313, 0.67309541194531818486015002203, 2.14348545924013479149918571766, 2.24496994235425324259953094475, 2.95222965365990008370619645064, 3.75244587907017742849056603563, 4.37541871248185439060241844236, 4.59940060664626995630977527685, 5.33840409146129457554554047888, 5.75348664454641273635208613074, 6.30745054784354111580924889557, 6.69354897610083707626986067754, 7.58171582189101263195110343002, 7.75318129817252636597864626224, 8.353678078345244855121736809398, 8.486584387266337560833474699133, 9.364117298900427358986604649774, 9.442535322036218021477214214738, 10.32201929136535336802421754363, 10.34219346670654745362660129872
