| L(s) = 1 | − 3·2-s − 5·3-s + 6·4-s − 2·5-s + 15·6-s + 3·7-s − 10·8-s + 10·9-s + 6·10-s − 3·11-s − 30·12-s − 9·14-s + 10·15-s + 15·16-s − 7·17-s − 30·18-s + 11·19-s − 12·20-s − 15·21-s + 9·22-s − 14·23-s + 50·24-s − 3·25-s − 6·27-s + 18·28-s + 2·29-s − 30·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.88·3-s + 3·4-s − 0.894·5-s + 6.12·6-s + 1.13·7-s − 3.53·8-s + 10/3·9-s + 1.89·10-s − 0.904·11-s − 8.66·12-s − 2.40·14-s + 2.58·15-s + 15/4·16-s − 1.69·17-s − 7.07·18-s + 2.52·19-s − 2.68·20-s − 3.27·21-s + 1.91·22-s − 2.91·23-s + 10.2·24-s − 3/5·25-s − 1.15·27-s + 3.40·28-s + 0.371·29-s − 5.47·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2963294160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2963294160\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + 5 T + 5 p T^{2} + 31 T^{3} + 5 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 12 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 53 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 7 T + 65 T^{2} + 245 T^{3} + 65 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 11 T + 53 T^{2} - 207 T^{3} + 53 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 14 T + 125 T^{2} + 700 T^{3} + 125 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 2 T + 79 T^{2} - 108 T^{3} + 79 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 14 T + 149 T^{2} - 924 T^{3} + 149 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 8 T + 123 T^{2} - 584 T^{3} + 123 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 9 T + 59 T^{2} - 5 p T^{3} + 59 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 19 T + 219 T^{2} - 1663 T^{3} + 219 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 14 T + 169 T^{2} + 1260 T^{3} + 169 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 14 T + 215 T^{2} + 1540 T^{3} + 215 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 9 T - 13 T^{2} + 745 T^{3} - 13 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 4 T + 123 T^{2} + 256 T^{3} + 123 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 3 T + 141 T^{2} + 151 T^{3} + 141 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 8 T + 169 T^{2} - 792 T^{3} + 169 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 13 T + 203 T^{2} - 1731 T^{3} + 203 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 16 T + 201 T^{2} + 2296 T^{3} + 201 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 21 T + 333 T^{2} - 3577 T^{3} + 333 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 25 T + 431 T^{2} - 4547 T^{3} + 431 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 31 T + 609 T^{2} - 7093 T^{3} + 609 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.914568448415514337049921090770, −7.82903960399757875216656571291, −7.60834012063641255809123566145, −7.48160656790992205306720460696, −6.76666597037889926788497029203, −6.66374413186232704310891696210, −6.41007816914957307570485983349, −6.10291964046230962642282979925, −5.95035556679516474524834357338, −5.88188541896118568314861053403, −5.31251146854386487314206177414, −5.20291576196928205554149800304, −4.92149114361521989162646915390, −4.41975582026212073702884079653, −4.38804252798376160650083964378, −4.05191347239270475198756712866, −3.42950504575877212774771127464, −3.04501669062144818976111592416, −2.68096277449697558890394705800, −2.14583860349101971017354626805, −2.07897276430616856690517221055, −1.49255767680470477826450064297, −0.66749074865546196207883589210, −0.65543476631712114573344777187, −0.53085694387015859920448286151,
0.53085694387015859920448286151, 0.65543476631712114573344777187, 0.66749074865546196207883589210, 1.49255767680470477826450064297, 2.07897276430616856690517221055, 2.14583860349101971017354626805, 2.68096277449697558890394705800, 3.04501669062144818976111592416, 3.42950504575877212774771127464, 4.05191347239270475198756712866, 4.38804252798376160650083964378, 4.41975582026212073702884079653, 4.92149114361521989162646915390, 5.20291576196928205554149800304, 5.31251146854386487314206177414, 5.88188541896118568314861053403, 5.95035556679516474524834357338, 6.10291964046230962642282979925, 6.41007816914957307570485983349, 6.66374413186232704310891696210, 6.76666597037889926788497029203, 7.48160656790992205306720460696, 7.60834012063641255809123566145, 7.82903960399757875216656571291, 7.914568448415514337049921090770
