| L(s) = 1 | − 4·5-s + 5·9-s − 14·11-s − 96·19-s − 109·25-s − 438·29-s + 596·31-s + 100·41-s − 20·45-s − 49·49-s + 56·55-s + 1.56e3·59-s + 976·61-s − 480·71-s + 2.13e3·79-s − 704·81-s − 1.21e3·89-s + 384·95-s − 70·99-s − 2.15e3·101-s + 742·109-s − 2.51e3·121-s + 936·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.357·5-s + 5/27·9-s − 0.383·11-s − 1.15·19-s − 0.871·25-s − 2.80·29-s + 3.45·31-s + 0.380·41-s − 0.0662·45-s − 1/7·49-s + 0.137·55-s + 3.45·59-s + 2.04·61-s − 0.802·71-s + 3.03·79-s − 0.965·81-s − 1.44·89-s + 0.414·95-s − 0.0710·99-s − 2.12·101-s + 0.652·109-s − 1.88·121-s + 0.669·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.510796993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510796993\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 7 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4385 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6105 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20970 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 219 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 298 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 72406 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 50 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 75242 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 190485 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 253654 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 782 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 p T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 357490 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 240 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 774670 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1065 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 70278 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 608 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 32425 T^{2} + p^{6} T^{4} \) |
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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
−13.31112254229149781426170379450, −12.51337370434276860726770174483, −11.76623672857565408910490739659, −11.50311602878014496273291315040, −11.02778203758305372183165033956, −10.17061610505598334835525397680, −10.06453247663004848722529240386, −9.345835668885093148366348389109, −8.694947501755633420481539429644, −7.968108529072243713872243392122, −7.943097375274716454694050123231, −6.88237955041895015158538067620, −6.59101155867001560165068144872, −5.68070084319586250504206583342, −5.22030927130237983889411941270, −4.12968050684881937610140351133, −3.97061752993602993476982166471, −2.74546756520459699321643646211, −1.99265987106281306028925924511, −0.60917697684599122741979120379,
0.60917697684599122741979120379, 1.99265987106281306028925924511, 2.74546756520459699321643646211, 3.97061752993602993476982166471, 4.12968050684881937610140351133, 5.22030927130237983889411941270, 5.68070084319586250504206583342, 6.59101155867001560165068144872, 6.88237955041895015158538067620, 7.943097375274716454694050123231, 7.968108529072243713872243392122, 8.694947501755633420481539429644, 9.345835668885093148366348389109, 10.06453247663004848722529240386, 10.17061610505598334835525397680, 11.02778203758305372183165033956, 11.50311602878014496273291315040, 11.76623672857565408910490739659, 12.51337370434276860726770174483, 13.31112254229149781426170379450
