| L(s) = 1 | + 3·2-s + 3·4-s + 6·5-s − 3·7-s + 18·10-s + 3·11-s − 3·13-s − 9·14-s − 3·16-s + 9·17-s − 3·19-s + 18·20-s + 9·22-s + 6·23-s + 12·25-s − 9·26-s − 9·28-s + 12·29-s − 12·31-s − 6·32-s + 27·34-s − 18·35-s − 3·37-s − 9·38-s − 3·41-s − 12·43-s + 9·44-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s + 2.68·5-s − 1.13·7-s + 5.69·10-s + 0.904·11-s − 0.832·13-s − 2.40·14-s − 3/4·16-s + 2.18·17-s − 0.688·19-s + 4.02·20-s + 1.91·22-s + 1.25·23-s + 12/5·25-s − 1.76·26-s − 1.70·28-s + 2.22·29-s − 2.15·31-s − 1.06·32-s + 4.63·34-s − 3.04·35-s − 0.493·37-s − 1.45·38-s − 0.468·41-s − 1.82·43-s + 1.35·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14348907 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14348907 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.041822512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.041822512\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 9 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 63 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 25 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 63 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 61 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 115 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 225 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 12 T + 114 T^{2} - 639 T^{3} + 114 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 763 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 87 T^{2} + 223 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 3 T + 69 T^{2} + 27 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1051 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 6 T + 78 T^{2} + 297 T^{3} + 78 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 18 T + 240 T^{2} - 1989 T^{3} + 240 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 21 T + 321 T^{2} + 2799 T^{3} + 321 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 785 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 695 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 9 T + 51 T^{2} + 279 T^{3} + 51 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 479 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 6 T + 186 T^{2} - 1001 T^{3} + 186 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 6 T + 222 T^{2} + 945 T^{3} + 222 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 78 T^{2} + 999 T^{3} + 78 p T^{4} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 15 T + 222 T^{2} - 2891 T^{3} + 222 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−10.62740667381714668091185561312, −10.44446913418299998873332249913, −10.13096140643082686485371177664, −10.01143109856155763317372996753, −9.489302751643217678452931395295, −9.322533113220758939042946339339, −9.250945483838276770216079088236, −8.587142896621799696795124404139, −8.289762168637513029507742100679, −7.69480844162426561675059666923, −7.09376223963121658702144740823, −6.73434674165298570847267662000, −6.69922177604601220464976850491, −5.97773236735291929112212846295, −5.80728829618544993629238015979, −5.76267162674386544228975097635, −5.05660061292174397537848008291, −4.86075960264171060465219091221, −4.73753752546484056772824916487, −3.77960377445360964289304377066, −3.47169965665101283778552642364, −3.24205600505402485305254034467, −2.57844828379127453678469692471, −1.95715542249369011083465327052, −1.41432758405549024508711089502,
1.41432758405549024508711089502, 1.95715542249369011083465327052, 2.57844828379127453678469692471, 3.24205600505402485305254034467, 3.47169965665101283778552642364, 3.77960377445360964289304377066, 4.73753752546484056772824916487, 4.86075960264171060465219091221, 5.05660061292174397537848008291, 5.76267162674386544228975097635, 5.80728829618544993629238015979, 5.97773236735291929112212846295, 6.69922177604601220464976850491, 6.73434674165298570847267662000, 7.09376223963121658702144740823, 7.69480844162426561675059666923, 8.289762168637513029507742100679, 8.587142896621799696795124404139, 9.250945483838276770216079088236, 9.322533113220758939042946339339, 9.489302751643217678452931395295, 10.01143109856155763317372996753, 10.13096140643082686485371177664, 10.44446913418299998873332249913, 10.62740667381714668091185561312
