| L(s) = 1 | − 3·4-s + 9·9-s + 10·11-s + 6·16-s + 2·19-s − 6·29-s + 8·31-s − 27·36-s + 34·41-s − 30·44-s + 10·49-s − 32·59-s + 4·61-s − 10·64-s + 16·71-s − 6·76-s + 34·81-s − 22·89-s + 90·99-s + 16·101-s + 12·109-s + 18·116-s + 25·121-s − 24·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 3·9-s + 3.01·11-s + 3/2·16-s + 0.458·19-s − 1.11·29-s + 1.43·31-s − 9/2·36-s + 5.30·41-s − 4.52·44-s + 10/7·49-s − 4.16·59-s + 0.512·61-s − 5/4·64-s + 1.89·71-s − 0.688·76-s + 34/9·81-s − 2.33·89-s + 9.04·99-s + 1.59·101-s + 1.14·109-s + 1.67·116-s + 2.27·121-s − 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.816328358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.816328358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 29 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 - p^{2} T^{2} + 47 T^{4} - 170 T^{6} + 47 p^{2} T^{8} - p^{6} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 10 T^{2} + 163 T^{4} - 964 T^{6} + 163 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 5 T + 25 T^{2} - 6 p T^{3} + 25 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 62 T^{2} + 1771 T^{4} - 29420 T^{6} + 1771 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 53 T^{2} + 1059 T^{4} - 15182 T^{6} + 1059 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - T + 53 T^{2} - 36 T^{3} + 53 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 6 T^{2} + 1235 T^{4} - 2972 T^{6} + 1235 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 4 T + 14 T^{2} - 90 T^{3} + 14 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 108 T^{2} + 6968 T^{4} - 317870 T^{6} + 6968 p^{2} T^{8} - 108 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 17 T + 195 T^{2} - 1392 T^{3} + 195 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 130 T^{2} + 10903 T^{4} - 550396 T^{6} + 10903 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 108 T^{2} + 6456 T^{4} - 278802 T^{6} + 6456 p^{2} T^{8} - 108 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 174 T^{2} + 14907 T^{4} - 899340 T^{6} + 14907 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 16 T + 238 T^{2} + 1956 T^{3} + 238 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 2 T + 76 T^{2} - 98 T^{3} + 76 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 291 T^{2} + 40370 T^{4} - 3383867 T^{6} + 40370 p^{2} T^{8} - 291 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 8 T + 179 T^{2} - 868 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 165 T^{2} + 11087 T^{4} - 539570 T^{6} + 11087 p^{2} T^{8} - 165 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 15 T^{2} + 432 T^{3} - 15 p T^{4} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 289 T^{2} + 40547 T^{4} - 3847590 T^{6} + 40547 p^{2} T^{8} - 289 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 11 T + 3 p T^{2} + 1800 T^{3} + 3 p^{2} T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 322 T^{2} + 57347 T^{4} - 6864372 T^{6} + 57347 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.71947617032199971105949779778, −4.66807970390785576666261146898, −4.65488623840619821175676546302, −4.64247614638952412323079602108, −4.23696167485628473193703507935, −4.18465263110509622299301245166, −4.02042723822584557214702077800, −3.97116776640865590787267829501, −3.91777118010830431631033251719, −3.82046933689122017144405633137, −3.39832252943560341707242469156, −3.27432813408693497642219305645, −3.24984839759305665718932998786, −2.79542467900995834716840911017, −2.61555166663799235648672990572, −2.56616887274346185010583350154, −2.11848267915247722735737454602, −2.07318508979689700033427751340, −1.66503505717602899396904290183, −1.44703607625611833275518825717, −1.39737108339364922699609699868, −1.12722578318187148976148965090, −0.899860982485450110058481782859, −0.853153442287446562542221416101, −0.38654334699099946757716324355,
0.38654334699099946757716324355, 0.853153442287446562542221416101, 0.899860982485450110058481782859, 1.12722578318187148976148965090, 1.39737108339364922699609699868, 1.44703607625611833275518825717, 1.66503505717602899396904290183, 2.07318508979689700033427751340, 2.11848267915247722735737454602, 2.56616887274346185010583350154, 2.61555166663799235648672990572, 2.79542467900995834716840911017, 3.24984839759305665718932998786, 3.27432813408693497642219305645, 3.39832252943560341707242469156, 3.82046933689122017144405633137, 3.91777118010830431631033251719, 3.97116776640865590787267829501, 4.02042723822584557214702077800, 4.18465263110509622299301245166, 4.23696167485628473193703507935, 4.64247614638952412323079602108, 4.65488623840619821175676546302, 4.66807970390785576666261146898, 4.71947617032199971105949779778
Plot not available for L-functions of degree greater than 10.
