| L(s) = 1 | + 3·3-s − 5-s + 7-s + 4·9-s − 2·11-s − 3·13-s − 3·15-s − 2·17-s − 2·19-s + 3·21-s + 10·23-s − 6·25-s + 6·27-s − 13·29-s + 3·31-s − 6·33-s − 35-s − 4·37-s − 9·39-s − 3·41-s − 13·43-s − 4·45-s + 12·47-s − 10·49-s − 6·51-s − 14·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 0.377·7-s + 4/3·9-s − 0.603·11-s − 0.832·13-s − 0.774·15-s − 0.485·17-s − 0.458·19-s + 0.654·21-s + 2.08·23-s − 6/5·25-s + 1.15·27-s − 2.41·29-s + 0.538·31-s − 1.04·33-s − 0.169·35-s − 0.657·37-s − 1.44·39-s − 0.468·41-s − 1.98·43-s − 0.596·45-s + 1.75·47-s − 1.42·49-s − 0.840·51-s − 1.92·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.245281867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.245281867\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 11 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 61 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 55 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 99 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 142 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 59 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 23 T + 271 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 134 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 25 T + 319 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 166 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.626909775185722067890891223447, −8.549336030735478096462910579120, −8.021162380852761823738338742854, −7.86808401180208209945294838866, −7.36868874061674564625614645022, −7.21029775945071738644406215484, −6.69235826602843930211618695574, −6.41368986094192785036220324573, −5.60310817646620517135580908330, −5.36292103305552409893719949211, −4.73601567000960388937364567316, −4.72792953074543892690514583190, −3.96994513869324872524694201751, −3.63894967963303536085836531378, −3.05430289660161308549977011408, −3.03635616245865865188449844245, −2.23475202435603698925073407177, −2.00742257113417754278105017779, −1.47461518742662741182570722894, −0.37741325867507487461208358011,
0.37741325867507487461208358011, 1.47461518742662741182570722894, 2.00742257113417754278105017779, 2.23475202435603698925073407177, 3.03635616245865865188449844245, 3.05430289660161308549977011408, 3.63894967963303536085836531378, 3.96994513869324872524694201751, 4.72792953074543892690514583190, 4.73601567000960388937364567316, 5.36292103305552409893719949211, 5.60310817646620517135580908330, 6.41368986094192785036220324573, 6.69235826602843930211618695574, 7.21029775945071738644406215484, 7.36868874061674564625614645022, 7.86808401180208209945294838866, 8.021162380852761823738338742854, 8.549336030735478096462910579120, 8.626909775185722067890891223447
