| L(s) = 1 | − 8·9-s − 2·16-s + 24·29-s + 24·31-s + 56·59-s − 32·61-s + 30·81-s + 56·89-s + 64·121-s + 127-s + 131-s + 137-s + 139-s + 16·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 8/3·9-s − 1/2·16-s + 4.45·29-s + 4.31·31-s + 7.29·59-s − 4.09·61-s + 10/3·81-s + 5.93·89-s + 5.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.57245587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.57245587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + T^{4} )^{2} \) |
| good | 11 | \( ( 1 - 32 T^{2} + 478 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 28 T^{4} + 52198 T^{8} - 28 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2}( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 - 48 T^{2} + 1118 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 508 T^{4} + 4678 T^{8} - 508 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( 1 - 604 T^{4} - 32474 T^{8} - 604 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 56 T^{2} + 3166 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 5444 T^{4} + 13734886 T^{8} + 5444 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 - 5788 T^{4} + 16086598 T^{8} - 5788 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 - 7708 T^{4} + 28156198 T^{8} - 7708 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 - 12988 T^{4} + 81854758 T^{8} - 12988 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 84 T^{2} + 8966 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 1148 T^{4} + 2279238 T^{8} - 1148 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 124 T^{2} + 11206 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 10436 T^{4} + 85152166 T^{8} + 10436 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 14 T + 222 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 21436 T^{4} + 227134086 T^{8} - 21436 p^{4} T^{12} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.13567091487969376383084119535, −4.12512731121220486731672086421, −4.11740365929026493087099668510, −3.92764146887418381231170706615, −3.77109264591087178469822730034, −3.39961482442963534351189283738, −3.22851630143862786286817971105, −3.19447162271887019175251545547, −3.16152314884113147663802019948, −3.05174984006961320401614955293, −2.98180971735644473254374030134, −2.74617374509289301774789604570, −2.73245663038668589666390815268, −2.50588024480223460835318361367, −2.21345645098514306089488111707, −2.17026125994278867033465355735, −2.01305572483798732988316843339, −1.98106037775045847767033871665, −1.78028967122714892201987358851, −1.33658826692241628706989022812, −0.948395692307427800136904344698, −0.802800001868085085622925658393, −0.69484391685693670569816982824, −0.69453161095106099646738360519, −0.50629594714646948518945950274,
0.50629594714646948518945950274, 0.69453161095106099646738360519, 0.69484391685693670569816982824, 0.802800001868085085622925658393, 0.948395692307427800136904344698, 1.33658826692241628706989022812, 1.78028967122714892201987358851, 1.98106037775045847767033871665, 2.01305572483798732988316843339, 2.17026125994278867033465355735, 2.21345645098514306089488111707, 2.50588024480223460835318361367, 2.73245663038668589666390815268, 2.74617374509289301774789604570, 2.98180971735644473254374030134, 3.05174984006961320401614955293, 3.16152314884113147663802019948, 3.19447162271887019175251545547, 3.22851630143862786286817971105, 3.39961482442963534351189283738, 3.77109264591087178469822730034, 3.92764146887418381231170706615, 4.11740365929026493087099668510, 4.12512731121220486731672086421, 4.13567091487969376383084119535
Plot not available for L-functions of degree greater than 10.
