| L(s) = 1 | − 6·13-s + 3·19-s − 9·27-s + 24·43-s + 39·61-s + 8·64-s + 15·67-s − 21·73-s + 39·79-s + 66·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 15·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.66·13-s + 0.688·19-s − 1.73·27-s + 3.65·43-s + 4.99·61-s + 64-s + 1.83·67-s − 2.45·73-s + 4.38·79-s + 6·121-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 − 1.15·169-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 + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.757099997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.757099997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{2} T^{3} + p^{3} T^{6} \) |
| 7 | \( 1 - 37 T^{3} + p^{3} T^{6} \) |
| 19 | \( ( 1 - T + p T^{2} )^{3} \) |
| good | 2 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 5 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) |
| 11 | \( ( 1 - p T^{2} )^{6} \) |
| 13 | \( ( 1 + 2 T + p T^{2} )^{3}( 1 + 19 T^{3} + p^{3} T^{6} ) \) |
| 17 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) |
| 23 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) |
| 29 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 289 T^{3} + p^{3} T^{6} )( 1 + 289 T^{3} + p^{3} T^{6} ) \) |
| 37 | \( ( 1 - 433 T^{3} + p^{3} T^{6} )( 1 + 433 T^{3} + p^{3} T^{6} ) \) |
| 41 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 8 T + p T^{2} )^{3}( 1 - 71 T^{3} + p^{3} T^{6} ) \) |
| 47 | \( 1 + p^{3} T^{6} + p^{6} T^{12} \) |
| 53 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 59 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 13 T + p T^{2} )^{3}( 1 + 719 T^{3} + p^{3} T^{6} ) \) |
| 67 | \( ( 1 - 5 T + p T^{2} )^{3}( 1 + 1007 T^{3} + p^{3} T^{6} ) \) |
| 71 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 73 | \( ( 1 + 7 T + p T^{2} )^{3}( 1 + 271 T^{3} + p^{3} T^{6} ) \) |
| 79 | \( ( 1 - 13 T + p T^{2} )^{3}( 1 + 1387 T^{3} + p^{3} T^{6} ) \) |
| 83 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{3} \) |
| 89 | \( 1 - p^{3} T^{6} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 1853 T^{3} + p^{3} T^{6} )( 1 - 523 T^{3} + p^{3} T^{6} ) \) |
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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
−5.87774866172489331286439027541, −5.78029872011096756921433388529, −5.72561555039769464736872031308, −5.61583700484902542248065144480, −5.56420083249721175401749412002, −5.24552120008214856527369992234, −5.11334658838148171892868099850, −4.64866226987951729075788626075, −4.58444903805445917878374321992, −4.55293315101465573128546993198, −4.44251918401123595887065048313, −3.91765208802344764865081506219, −3.70783929402503511932717339759, −3.67222805933512045424176546899, −3.62044400889006949829225446470, −3.16977595993858700300513351497, −2.99474643768330642808540544226, −2.53981430110923004268519232776, −2.44973896291346637507826679191, −2.16291786480285047174298181421, −2.04683656357540774098557339537, −1.96065256592125456763559193113, −1.13898267652173968359687872981, −0.789694901505097848484307887486, −0.59358031708113215263733912654,
0.59358031708113215263733912654, 0.789694901505097848484307887486, 1.13898267652173968359687872981, 1.96065256592125456763559193113, 2.04683656357540774098557339537, 2.16291786480285047174298181421, 2.44973896291346637507826679191, 2.53981430110923004268519232776, 2.99474643768330642808540544226, 3.16977595993858700300513351497, 3.62044400889006949829225446470, 3.67222805933512045424176546899, 3.70783929402503511932717339759, 3.91765208802344764865081506219, 4.44251918401123595887065048313, 4.55293315101465573128546993198, 4.58444903805445917878374321992, 4.64866226987951729075788626075, 5.11334658838148171892868099850, 5.24552120008214856527369992234, 5.56420083249721175401749412002, 5.61583700484902542248065144480, 5.72561555039769464736872031308, 5.78029872011096756921433388529, 5.87774866172489331286439027541
Plot not available for L-functions of degree greater than 10.
