| L(s) = 1 | − 3·4-s − 9-s + 6·11-s + 6·16-s − 6·19-s + 8·29-s − 10·31-s + 3·36-s + 2·41-s − 18·44-s − 9·49-s − 28·59-s + 2·61-s − 10·64-s + 22·71-s + 18·76-s + 8·79-s − 11·81-s − 36·89-s − 6·99-s − 8·101-s + 30·109-s − 24·116-s + 39·121-s + 30·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1/3·9-s + 1.80·11-s + 3/2·16-s − 1.37·19-s + 1.48·29-s − 1.79·31-s + 1/2·36-s + 0.312·41-s − 2.71·44-s − 9/7·49-s − 3.64·59-s + 0.256·61-s − 5/4·64-s + 2.61·71-s + 2.06·76-s + 0.900·79-s − 1.22·81-s − 3.81·89-s − 0.603·99-s − 0.796·101-s + 2.87·109-s − 2.22·116-s + 3.54·121-s + 2.69·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223573203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223573203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 23 | \( ( 1 + T^{2} )^{3} \) |
| good | 3 | \( 1 + T^{2} + 4 p T^{4} + 4 p^{3} T^{8} + p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 + 9 T^{2} + 132 T^{4} + 680 T^{6} + 132 p^{2} T^{8} + 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 3 T - 6 T^{2} + 78 T^{3} - 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 47 T^{2} + 1184 T^{4} - 18968 T^{6} + 1184 p^{2} T^{8} - 47 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 67 T^{2} + 2256 T^{4} - 47480 T^{6} + 2256 p^{2} T^{8} - 67 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 3 T + 36 T^{2} + 50 T^{3} + 36 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 4 T + 55 T^{2} - 256 T^{3} + 55 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 5 T + 86 T^{2} + 302 T^{3} + 86 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 138 T^{2} + 9831 T^{4} - 449932 T^{6} + 9831 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - T + 64 T^{2} + 104 T^{3} + 64 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{3} \) |
| 47 | \( 1 - 2 p T^{2} + 5871 T^{4} - 261284 T^{6} + 5871 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{3} \) |
| 59 | \( ( 1 + 14 T + 205 T^{2} + 1508 T^{3} + 205 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - T + 26 T^{2} + 404 T^{3} + 26 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 50 T^{2} - 121 T^{4} + 11044 T^{6} - 121 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 11 T + 244 T^{2} - 1586 T^{3} + 244 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 294 T^{2} + 43455 T^{4} - 3927508 T^{6} + 43455 p^{2} T^{8} - 294 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 4 T - 3 T^{2} + 520 T^{3} - 3 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 394 T^{2} + 70743 T^{4} - 7449164 T^{6} + 70743 p^{2} T^{8} - 394 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 18 T + 219 T^{2} + 2052 T^{3} + 219 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 51 T^{2} + 1992 T^{4} - 1224520 T^{6} + 1992 p^{2} T^{8} - 51 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.21094734394524584836971853641, −4.80008888994289150010264806741, −4.76459591049384446159368323857, −4.66043827352412813309012107324, −4.49262621326057561544454997090, −4.38582424447294994261595384175, −4.31997709255133834035199291930, −4.11973678190586253740737030984, −3.76810244846693806199911413662, −3.70426286237112574043637927993, −3.67210698886093732212753899882, −3.37302478685458703711872654179, −3.13956573974367777269258061381, −3.01182051812716714310107235149, −2.96239023006359401949048792511, −2.62031034995037913229378134318, −2.33734163032712120420745458619, −2.12572910393576819568297270070, −1.79891088811823884226094515511, −1.64778265089839964617259187778, −1.55448698628557568965985727263, −1.29316392511902349318088390131, −0.71884421102320223697660578416, −0.69254865095357059177978443704, −0.21796593097137714526795225984,
0.21796593097137714526795225984, 0.69254865095357059177978443704, 0.71884421102320223697660578416, 1.29316392511902349318088390131, 1.55448698628557568965985727263, 1.64778265089839964617259187778, 1.79891088811823884226094515511, 2.12572910393576819568297270070, 2.33734163032712120420745458619, 2.62031034995037913229378134318, 2.96239023006359401949048792511, 3.01182051812716714310107235149, 3.13956573974367777269258061381, 3.37302478685458703711872654179, 3.67210698886093732212753899882, 3.70426286237112574043637927993, 3.76810244846693806199911413662, 4.11973678190586253740737030984, 4.31997709255133834035199291930, 4.38582424447294994261595384175, 4.49262621326057561544454997090, 4.66043827352412813309012107324, 4.76459591049384446159368323857, 4.80008888994289150010264806741, 5.21094734394524584836971853641
Plot not available for L-functions of degree greater than 10.
