| L(s) = 1 | + 2·3-s + 3·5-s − 9-s − 5·11-s − 2·13-s + 6·15-s − 10·17-s − 3·19-s − 17·23-s − 25-s − 6·27-s + 2·29-s − 6·31-s − 10·33-s + 14·37-s − 4·39-s − 4·41-s − 17·43-s − 3·45-s + 9·47-s − 20·51-s − 22·53-s − 15·55-s − 6·57-s − 6·59-s − 61-s − 6·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.34·5-s − 1/3·9-s − 1.50·11-s − 0.554·13-s + 1.54·15-s − 2.42·17-s − 0.688·19-s − 3.54·23-s − 1/5·25-s − 1.15·27-s + 0.371·29-s − 1.07·31-s − 1.74·33-s + 2.30·37-s − 0.640·39-s − 0.624·41-s − 2.59·43-s − 0.447·45-s + 1.31·47-s − 2.80·51-s − 3.02·53-s − 2.02·55-s − 0.794·57-s − 0.781·59-s − 0.128·61-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 3 T + 2 p T^{2} - 19 T^{3} + 2 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5 T + 34 T^{2} + 109 T^{3} + 34 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 33 T^{2} + 54 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 10 T + 67 T^{2} + 304 T^{3} + 67 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 17 T + 160 T^{2} + 931 T^{3} + 160 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} - 134 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 57 T^{2} + 358 T^{3} + 57 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 14 T + 169 T^{2} - 30 p T^{3} + 169 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 51 T^{2} + 472 T^{3} + 51 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T + 220 T^{2} + 1611 T^{3} + 220 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 9 T + 124 T^{2} - 735 T^{3} + 124 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 22 T + 225 T^{2} + 1706 T^{3} + 225 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 61 | $S_4\times C_2$ | \( 1 + T + 126 T^{2} + 261 T^{3} + 126 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 133 T^{2} - 100 T^{3} + 133 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 18 T + 277 T^{2} + 2514 T^{3} + 277 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 21 T + 334 T^{2} - 3217 T^{3} + 334 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 253 T^{2} - 1266 T^{3} + 253 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 27 T + 426 T^{2} + 4439 T^{3} + 426 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 30 T + 501 T^{2} - 5502 T^{3} + 501 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 119 T^{2} + 1252 T^{3} + 119 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.099669021231709156958361399414, −7.62795009869192896599743433183, −7.60230793182888894231565949875, −7.30681525802612281388261918008, −6.96253551361268373923940179101, −6.43222414792494547575130884634, −6.40064348689333503799455181735, −6.06538119631104881468455499311, −6.03762561085899826738023943401, −5.86789917482284339606472000489, −5.40641295031324779674898730289, −5.05848951606485923657501294482, −4.83534027009847245266282219790, −4.54651051521495809708834312671, −4.45480908886158491450415062604, −3.94133713281796765685396976345, −3.59944191780398294406207318495, −3.54144210120259368279489682961, −3.00543222224667717052983590157, −2.51580343142356178859126788996, −2.43915757792854877372342510325, −2.35042024607911817423970034714, −1.93839213548530716833934848952, −1.80714947930969138668345858766, −1.33680118005179253749863889804, 0, 0, 0,
1.33680118005179253749863889804, 1.80714947930969138668345858766, 1.93839213548530716833934848952, 2.35042024607911817423970034714, 2.43915757792854877372342510325, 2.51580343142356178859126788996, 3.00543222224667717052983590157, 3.54144210120259368279489682961, 3.59944191780398294406207318495, 3.94133713281796765685396976345, 4.45480908886158491450415062604, 4.54651051521495809708834312671, 4.83534027009847245266282219790, 5.05848951606485923657501294482, 5.40641295031324779674898730289, 5.86789917482284339606472000489, 6.03762561085899826738023943401, 6.06538119631104881468455499311, 6.40064348689333503799455181735, 6.43222414792494547575130884634, 6.96253551361268373923940179101, 7.30681525802612281388261918008, 7.60230793182888894231565949875, 7.62795009869192896599743433183, 8.099669021231709156958361399414
