| L(s) = 1 | + 3·3-s − 6·11-s + 3·13-s − 3·17-s + 9·19-s + 3·23-s − 3·25-s − 12·27-s − 12·29-s + 12·31-s − 18·33-s − 21·37-s + 9·39-s + 3·41-s − 3·43-s + 3·47-s − 9·51-s + 18·53-s + 27·57-s + 6·59-s + 30·61-s − 18·67-s + 9·69-s + 12·71-s + 12·73-s − 9·75-s + 12·79-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.80·11-s + 0.832·13-s − 0.727·17-s + 2.06·19-s + 0.625·23-s − 3/5·25-s − 2.30·27-s − 2.22·29-s + 2.15·31-s − 3.13·33-s − 3.45·37-s + 1.44·39-s + 0.468·41-s − 0.457·43-s + 0.437·47-s − 1.26·51-s + 2.47·53-s + 3.57·57-s + 0.781·59-s + 3.84·61-s − 2.19·67-s + 1.08·69-s + 1.42·71-s + 1.40·73-s − 1.03·75-s + 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.220782746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.220782746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $A_4\times C_2$ | \( 1 - p T + p^{2} T^{2} - 5 p T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 3 T^{2} + 8 T^{3} + 3 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 6 T + 9 T^{2} - 4 T^{3} + 9 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 21 T^{2} - 21 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 99 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 9 T + 63 T^{2} - 269 T^{3} + 63 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 101 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 12 T + 123 T^{2} + 704 T^{3} + 123 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 552 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 21 T + 237 T^{2} + 1733 T^{3} + 237 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 3 T + 93 T^{2} + 239 T^{3} + 93 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 51 T^{2} + 99 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 18 T + 231 T^{2} - 1980 T^{3} + 231 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 141 T^{2} - 556 T^{3} + 141 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 30 T + 471 T^{2} - 4532 T^{3} + 471 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 18 T + 261 T^{2} + 2276 T^{3} + 261 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 12 T + 177 T^{2} - 1728 T^{3} + 177 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 1688 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 12 T + 177 T^{2} - 1744 T^{3} + 177 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 18 T + 309 T^{2} - 2852 T^{3} + 309 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 3 T + 153 T^{2} - 265 T^{3} + 153 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 3 T + 231 T^{2} + 349 T^{3} + 231 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.01680244584457631112103202618, −6.86452946889045734278285942163, −6.59554958731342173921283181718, −6.30065719283655566542788805565, −5.74833831602528269730088178406, −5.57557656522900925653650236892, −5.52586420916382386165309134503, −5.28493063658316600960156615118, −5.23200476854837271374092992326, −4.95425904708239323460192277839, −4.36465833139029406599992819016, −4.21930090876249853315009267103, −3.86463085249833848293347314339, −3.53322516311942860060558968961, −3.50082659386873857174520808414, −3.28731812069518168918394301746, −2.89032812791184590673560434094, −2.65574194162699490780259720116, −2.58763324687183055811830306112, −2.03223794366855792545218592851, −1.89558360080865830076153663568, −1.85677240800472376910598874370, −0.995098511673670613578431863761, −0.61638799082704095759805039630, −0.47039839218349477654940827214,
0.47039839218349477654940827214, 0.61638799082704095759805039630, 0.995098511673670613578431863761, 1.85677240800472376910598874370, 1.89558360080865830076153663568, 2.03223794366855792545218592851, 2.58763324687183055811830306112, 2.65574194162699490780259720116, 2.89032812791184590673560434094, 3.28731812069518168918394301746, 3.50082659386873857174520808414, 3.53322516311942860060558968961, 3.86463085249833848293347314339, 4.21930090876249853315009267103, 4.36465833139029406599992819016, 4.95425904708239323460192277839, 5.23200476854837271374092992326, 5.28493063658316600960156615118, 5.52586420916382386165309134503, 5.57557656522900925653650236892, 5.74833831602528269730088178406, 6.30065719283655566542788805565, 6.59554958731342173921283181718, 6.86452946889045734278285942163, 7.01680244584457631112103202618
