| L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 3·5-s + 6·6-s − 3·7-s − 2·8-s + 6·9-s + 6·10-s − 2·11-s − 3·12-s − 3·13-s + 6·14-s + 9·15-s + 7·16-s − 8·17-s − 12·18-s − 7·19-s − 3·20-s + 9·21-s + 4·22-s − 9·23-s + 6·24-s − 2·25-s + 6·26-s − 10·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s − 1.13·7-s − 0.707·8-s + 2·9-s + 1.89·10-s − 0.603·11-s − 0.866·12-s − 0.832·13-s + 1.60·14-s + 2.32·15-s + 7/4·16-s − 1.94·17-s − 2.82·18-s − 1.60·19-s − 0.670·20-s + 1.96·21-s + 0.852·22-s − 1.87·23-s + 1.22·24-s − 2/5·25-s + 1.17·26-s − 1.92·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20346417 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20346417 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 + p T + 3 T^{2} + 3 p T^{3} + 3 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 3 T + 11 T^{2} + 22 T^{3} + 11 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $D_{6}$ | \( 1 + 2 T + 5 T^{2} + 52 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 55 T^{2} + 240 T^{3} + 55 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 7 T + 41 T^{2} + 138 T^{3} + 41 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 89 T^{2} + 422 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + T + 55 T^{2} - 18 T^{3} + 55 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 7 T + 53 T^{2} + 162 T^{3} + 53 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 12 T + 131 T^{2} - 856 T^{3} + 131 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 139 T^{2} + 804 T^{3} + 139 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + T + 113 T^{2} + 102 T^{3} + 113 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 17 T + 221 T^{2} + 1666 T^{3} + 221 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 5 T + 63 T^{2} + 678 T^{3} + 63 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 197 T^{2} + 1352 T^{3} + 197 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 147 T^{2} - 988 T^{3} + 147 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 73 T^{2} + 340 T^{3} + 73 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 121 T^{2} + 72 T^{3} + 121 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 5 T + 75 T^{2} + 1166 T^{3} + 75 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 13 T + 277 T^{2} + 2086 T^{3} + 277 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + T + 217 T^{2} + 90 T^{3} + 217 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 13 T + 263 T^{2} + 1970 T^{3} + 263 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 9 T + 107 T^{2} + 1222 T^{3} + 107 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−11.20218987386593940626435282094, −10.94997439974829730867424835092, −10.32685444937641283860534458383, −10.06422683908761719776712251338, −9.959612607198591943917868638236, −9.516730262353179719824907430644, −9.511506926394470013522263309264, −8.664501218032730731758554958692, −8.555876686371260856322866158037, −8.472100238035409279247066236254, −7.69501212524816592898004382079, −7.64935459807238590060030704132, −7.26950254812755252054339121657, −6.76895350317749116357626020143, −6.31248303577370789161756290083, −6.24162795580474036632500947690, −5.96638571999111979418595354651, −5.34206926517614187399937750113, −4.95317750032561997593405383043, −4.31610652378245999110681454436, −4.22352194995039972389463480385, −3.68583120448531760208184043786, −3.17105547427276418647900894383, −2.32837515651647384683207389232, −1.80095072107294184984518087908, 0, 0, 0,
1.80095072107294184984518087908, 2.32837515651647384683207389232, 3.17105547427276418647900894383, 3.68583120448531760208184043786, 4.22352194995039972389463480385, 4.31610652378245999110681454436, 4.95317750032561997593405383043, 5.34206926517614187399937750113, 5.96638571999111979418595354651, 6.24162795580474036632500947690, 6.31248303577370789161756290083, 6.76895350317749116357626020143, 7.26950254812755252054339121657, 7.64935459807238590060030704132, 7.69501212524816592898004382079, 8.472100238035409279247066236254, 8.555876686371260856322866158037, 8.664501218032730731758554958692, 9.511506926394470013522263309264, 9.516730262353179719824907430644, 9.959612607198591943917868638236, 10.06422683908761719776712251338, 10.32685444937641283860534458383, 10.94997439974829730867424835092, 11.20218987386593940626435282094
