L(s) = 1 | + 9-s + 16·13-s + 8·25-s − 20·37-s − 60·61-s − 28·73-s − 3·81-s + 4·97-s + 28·109-s + 16·117-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | + 1/3·9-s + 4.43·13-s + 8/5·25-s − 3.28·37-s − 7.68·61-s − 3.27·73-s − 1/3·81-s + 0.406·97-s + 2.68·109-s + 1.47·117-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.612232187\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.612232187\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T^{2} + 4 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
good | 5 | \( ( 1 - 4 T^{2} - 3 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} + 189 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 25 T^{2} + 720 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 65 T^{2} + 1764 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 79 T^{2} + 2604 T^{4} + 79 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 31 T^{2} + 1908 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 11 T + p T^{2} )^{4}( 1 + 11 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 + 5 T + 66 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 64 T^{2} + 2334 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 80 T^{2} + 3246 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 71 T^{2} + 4524 T^{4} + 71 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 20 T^{2} + 5205 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 76 T^{2} + 7893 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - T^{2} - 6540 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 160 T^{2} + 14430 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 7 T + 144 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 17 T + p T^{2} )^{4}( 1 + 17 T + p T^{2} )^{4} \) |
| 83 | \( ( 1 + 301 T^{2} + 36300 T^{4} + 301 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 109 T^{2} + 13668 T^{4} - 109 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - T + 180 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.89810496769573762066433373480, −3.57509082069159517739135763014, −3.46579602862620555326337417145, −3.32333587827040788653209079650, −3.18873377947031595613302080698, −3.12681993498247082063116349885, −3.11966522623181070840271630150, −3.05926442212231722707375156643, −2.95627693434053131161675613289, −2.70157624477866184756477434297, −2.45109175512055151720688752235, −2.37339766978233231975006619838, −2.37225419229712614514358576783, −1.90464197320030347645127945763, −1.79797490420952672627427517993, −1.66139833647787902068473503064, −1.62005140829730079617523043769, −1.60943426078595621445160800176, −1.27058218406049754620280536191, −1.24757270643941948072164164676, −1.10540343999697545810086469301, −1.08638968794386640613120141256, −0.55833128485709130910077514514, −0.30759793058353097412479755842, −0.27205766481179060776121652906,
0.27205766481179060776121652906, 0.30759793058353097412479755842, 0.55833128485709130910077514514, 1.08638968794386640613120141256, 1.10540343999697545810086469301, 1.24757270643941948072164164676, 1.27058218406049754620280536191, 1.60943426078595621445160800176, 1.62005140829730079617523043769, 1.66139833647787902068473503064, 1.79797490420952672627427517993, 1.90464197320030347645127945763, 2.37225419229712614514358576783, 2.37339766978233231975006619838, 2.45109175512055151720688752235, 2.70157624477866184756477434297, 2.95627693434053131161675613289, 3.05926442212231722707375156643, 3.11966522623181070840271630150, 3.12681993498247082063116349885, 3.18873377947031595613302080698, 3.32333587827040788653209079650, 3.46579602862620555326337417145, 3.57509082069159517739135763014, 3.89810496769573762066433373480
Plot not available for L-functions of degree greater than 10.