| L(s) = 1 | − 4·4-s + 38·9-s + 10·11-s + 16·16-s + 84·19-s − 2·29-s + 452·31-s − 152·36-s + 32·41-s − 40·44-s − 49·49-s − 212·59-s + 96·61-s − 64·64-s − 30·71-s − 336·76-s + 2.50e3·79-s + 715·81-s − 172·89-s + 380·99-s + 3.65e3·109-s + 8·116-s − 2.58e3·121-s − 1.80e3·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.40·9-s + 0.274·11-s + 1/4·16-s + 1.01·19-s − 0.0128·29-s + 2.61·31-s − 0.703·36-s + 0.121·41-s − 0.137·44-s − 1/7·49-s − 0.467·59-s + 0.201·61-s − 1/8·64-s − 0.0501·71-s − 0.507·76-s + 3.56·79-s + 0.980·81-s − 0.204·89-s + 0.385·99-s + 3.21·109-s + 0.00640·116-s − 1.94·121-s − 1.30·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.077620461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.077620461\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2330 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9682 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 42 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6291 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 226 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100945 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 80053 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 96090 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 139350 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 106 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 368237 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 311902 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1253 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 569010 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 86 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1321246 T^{2} + p^{6} T^{4} \) |
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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
−11.38989181369026411067706938860, −10.66560709921042268840084654607, −10.26925474773874803237329613024, −9.948296661772883470218523104855, −9.330204703659582403276758708952, −9.290509715300972407834825053010, −8.232379886272797760183249487025, −8.228567947035707006389265355201, −7.39117894098126002599413150119, −7.16949694773755817967264224995, −6.26373415691182703731166237286, −6.25705035058007297310911286034, −5.13235829345022881097627823780, −4.86745854549275649088650486958, −4.28160841356787353520963141142, −3.71589610073648110915943104635, −3.05792702224212663854573183874, −2.18692095830498861440865461940, −1.24784989889817536008594891379, −0.72845727608546573999796201431,
0.72845727608546573999796201431, 1.24784989889817536008594891379, 2.18692095830498861440865461940, 3.05792702224212663854573183874, 3.71589610073648110915943104635, 4.28160841356787353520963141142, 4.86745854549275649088650486958, 5.13235829345022881097627823780, 6.25705035058007297310911286034, 6.26373415691182703731166237286, 7.16949694773755817967264224995, 7.39117894098126002599413150119, 8.228567947035707006389265355201, 8.232379886272797760183249487025, 9.290509715300972407834825053010, 9.330204703659582403276758708952, 9.948296661772883470218523104855, 10.26925474773874803237329613024, 10.66560709921042268840084654607, 11.38989181369026411067706938860
