| L(s) = 1 | − 3·4-s − 10·5-s + 2·7-s + 12·9-s − 4·13-s + 6·16-s + 30·20-s + 33·25-s − 6·28-s − 20·35-s − 36·36-s − 120·45-s − 21·49-s + 12·52-s + 48·53-s − 38·59-s + 24·63-s − 10·64-s + 40·65-s + 66·67-s − 2·71-s − 60·80-s + 76·81-s − 28·83-s − 8·91-s − 99·100-s − 22·103-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 4.47·5-s + 0.755·7-s + 4·9-s − 1.10·13-s + 3/2·16-s + 6.70·20-s + 33/5·25-s − 1.13·28-s − 3.38·35-s − 6·36-s − 17.8·45-s − 3·49-s + 1.66·52-s + 6.59·53-s − 4.94·59-s + 3.02·63-s − 5/4·64-s + 4.96·65-s + 8.06·67-s − 0.237·71-s − 6.70·80-s + 76/9·81-s − 3.07·83-s − 0.838·91-s − 9.89·100-s − 2.16·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.327877758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327877758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 4 p T^{2} + 68 T^{4} - 245 T^{6} + 68 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 5 | \( ( 1 + p T + 21 T^{2} + 51 T^{3} + 21 p T^{4} + p^{3} T^{5} + p^{3} T^{6} )^{2} \) |
| 7 | \( ( 1 - T + 12 T^{2} - 13 T^{3} + 12 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 21 T^{2} + 321 T^{4} - 3913 T^{6} + 321 p^{2} T^{8} - 21 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 2 T + 24 T^{2} + 23 T^{3} + 24 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 61 T^{2} + 2021 T^{4} - 41601 T^{6} + 2021 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 48 T^{2} + 1508 T^{4} - 34685 T^{6} + 1508 p^{2} T^{8} - 48 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 20 T^{2} - 91 T^{3} + 20 p T^{4} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 89 T^{2} + 3253 T^{4} - 88361 T^{6} + 3253 p^{2} T^{8} - 89 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 121 T^{2} + 8537 T^{4} - 382425 T^{6} + 8537 p^{2} T^{8} - 121 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 136 T^{2} + 8532 T^{4} - 380233 T^{6} + 8532 p^{2} T^{8} - 136 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 69 T^{2} + 2577 T^{4} - 84265 T^{6} + 2577 p^{2} T^{8} - 69 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 276 T^{2} + 32012 T^{4} - 1997405 T^{6} + 32012 p^{2} T^{8} - 276 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 24 T + 330 T^{2} - 2881 T^{3} + 330 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( ( 1 + 19 T + 281 T^{2} + 2411 T^{3} + 281 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( 1 - 164 T^{2} + 16276 T^{4} - 1218977 T^{6} + 16276 p^{2} T^{8} - 164 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 11 T + p T^{2} )^{6} \) |
| 71 | \( ( 1 + T + 127 T^{2} + 393 T^{3} + 127 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 244 T^{2} + 35368 T^{4} - 3100129 T^{6} + 35368 p^{2} T^{8} - 244 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( 1 - 390 T^{2} + 68639 T^{4} - 6959924 T^{6} + 68639 p^{2} T^{8} - 390 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 14 T + 200 T^{2} + 1435 T^{3} + 200 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 - 307 T^{2} + 52454 T^{4} - 5630247 T^{6} + 52454 p^{2} T^{8} - 307 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 446 T^{2} + 91135 T^{4} - 11101892 T^{6} + 91135 p^{2} T^{8} - 446 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.76076385973424664058487940249, −4.70496108697588100248357490403, −4.65679162448887597001138913754, −4.20229760804423542414947549777, −4.14501594889992040763418085927, −4.11320630761868050333526073678, −3.94838263521746159857859179632, −3.89300131853483799409977260790, −3.89029765249853223932086383630, −3.63766722797520961383953660228, −3.54715792852707993800563610104, −3.51963735372079627967107268878, −2.91426178417691687952594990538, −2.82248892052248904756786079167, −2.80272840119542404614576301137, −2.41381723270283214485143575915, −2.10691031206098564145246849339, −1.83656458651923032135646931967, −1.74482754018073685601675560650, −1.47627116394324032232782241105, −1.42541471130612851102772138537, −0.879397408550587952784731948560, −0.60142126382261707829141015958, −0.51393090513116417932601581206, −0.32491318083202330320569179919,
0.32491318083202330320569179919, 0.51393090513116417932601581206, 0.60142126382261707829141015958, 0.879397408550587952784731948560, 1.42541471130612851102772138537, 1.47627116394324032232782241105, 1.74482754018073685601675560650, 1.83656458651923032135646931967, 2.10691031206098564145246849339, 2.41381723270283214485143575915, 2.80272840119542404614576301137, 2.82248892052248904756786079167, 2.91426178417691687952594990538, 3.51963735372079627967107268878, 3.54715792852707993800563610104, 3.63766722797520961383953660228, 3.89029765249853223932086383630, 3.89300131853483799409977260790, 3.94838263521746159857859179632, 4.11320630761868050333526073678, 4.14501594889992040763418085927, 4.20229760804423542414947549777, 4.65679162448887597001138913754, 4.70496108697588100248357490403, 4.76076385973424664058487940249
Plot not available for L-functions of degree greater than 10.
