| L(s) = 1 | − 3·4-s − 3·9-s + 6·11-s + 6·16-s + 2·19-s + 6·29-s + 36·31-s + 9·36-s + 38·41-s − 18·44-s + 11·49-s + 40·61-s − 10·64-s + 24·71-s − 6·76-s + 6·79-s + 6·81-s − 16·89-s − 18·99-s + 16·101-s − 20·109-s − 18·116-s + 25·121-s − 108·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 9-s + 1.80·11-s + 3/2·16-s + 0.458·19-s + 1.11·29-s + 6.46·31-s + 3/2·36-s + 5.93·41-s − 2.71·44-s + 11/7·49-s + 5.12·61-s − 5/4·64-s + 2.84·71-s − 0.688·76-s + 0.675·79-s + 2/3·81-s − 1.69·89-s − 1.80·99-s + 1.59·101-s − 1.91·109-s − 1.67·116-s + 2.27·121-s − 9.69·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{12} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.21711539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.21711539\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 23 | \( ( 1 + T^{2} )^{3} \) |
| good | 7 | \( 1 - 11 T^{2} + 114 T^{4} - 1083 T^{6} + 114 p^{2} T^{8} - 11 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 3 T + T^{2} + 34 T^{3} + p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - p T^{2} + 155 T^{4} - 3886 T^{6} + 155 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 64 T^{2} + 2200 T^{4} - 46034 T^{6} + 2200 p^{2} T^{8} - 64 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - T + 25 T^{2} - 50 T^{3} + 25 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - T + p T^{2} )^{6} \) |
| 31 | \( ( 1 - 6 T + p T^{2} )^{6} \) |
| 37 | \( 1 - 184 T^{2} + 15360 T^{4} - 732554 T^{6} + 15360 p^{2} T^{8} - 184 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 19 T + 211 T^{2} - 1630 T^{3} + 211 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 137 T^{2} + 8435 T^{4} - 376502 T^{6} + 8435 p^{2} T^{8} - 137 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 196 T^{2} + 18528 T^{4} - 1075250 T^{6} + 18528 p^{2} T^{8} - 196 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 138 T^{2} + 14215 T^{4} - 842636 T^{6} + 14215 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 37 T^{2} + 528 T^{3} + 37 p T^{4} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 20 T + 255 T^{2} - 2280 T^{3} + 255 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 138 T^{2} + 14775 T^{4} - 1218508 T^{6} + 14775 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 12 T + 226 T^{2} - 1694 T^{3} + 226 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 315 T^{2} + 47686 T^{4} - 4382191 T^{6} + 47686 p^{2} T^{8} - 315 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 3 T + 205 T^{2} - 374 T^{3} + 205 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 147 T^{2} + 12522 T^{4} - 671235 T^{6} + 12522 p^{2} T^{8} - 147 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 8 T + 270 T^{2} + 1366 T^{3} + 270 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 426 T^{2} + 83791 T^{4} - 10049932 T^{6} + 83791 p^{2} T^{8} - 426 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.37012489574141527862440798786, −4.28062574055112231648766488315, −4.22745965249450570205597122552, −4.22144729487272665953416126738, −3.74715677100648300085133051539, −3.74499142717786684657528683130, −3.61273501803460507173150813844, −3.60069268673159444913327506827, −3.32853801070232468896129159873, −3.15551800638952237678470667704, −2.64509152558798713703371654432, −2.62831816952794226005636802198, −2.61545389373413665461971020387, −2.57538469926081953333294082133, −2.53517132950387160586888312013, −2.40410188527741793388365319494, −1.95797266134809035581235808402, −1.66846619800923977485071334795, −1.46684015314601679767745044240, −1.03490036169392673534131713959, −1.00607036885451214413473720702, −0.869151785701531387695228366154, −0.815802400370177258680396480171, −0.67511086504351978830107174705, −0.42272019206909735018501617864,
0.42272019206909735018501617864, 0.67511086504351978830107174705, 0.815802400370177258680396480171, 0.869151785701531387695228366154, 1.00607036885451214413473720702, 1.03490036169392673534131713959, 1.46684015314601679767745044240, 1.66846619800923977485071334795, 1.95797266134809035581235808402, 2.40410188527741793388365319494, 2.53517132950387160586888312013, 2.57538469926081953333294082133, 2.61545389373413665461971020387, 2.62831816952794226005636802198, 2.64509152558798713703371654432, 3.15551800638952237678470667704, 3.32853801070232468896129159873, 3.60069268673159444913327506827, 3.61273501803460507173150813844, 3.74499142717786684657528683130, 3.74715677100648300085133051539, 4.22144729487272665953416126738, 4.22745965249450570205597122552, 4.28062574055112231648766488315, 4.37012489574141527862440798786
Plot not available for L-functions of degree greater than 10.
