| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s + 6·7-s − 4·8-s + 6·11-s + 6·12-s − 12·14-s + 5·16-s + 6·17-s + 6·19-s + 12·21-s − 12·22-s + 12·23-s − 8·24-s − 2·27-s + 18·28-s − 12·29-s + 6·31-s − 6·32-s + 12·33-s − 12·34-s + 12·37-s − 12·38-s − 24·42-s + 4·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s + 2.26·7-s − 1.41·8-s + 1.80·11-s + 1.73·12-s − 3.20·14-s + 5/4·16-s + 1.45·17-s + 1.37·19-s + 2.61·21-s − 2.55·22-s + 2.50·23-s − 1.63·24-s − 0.384·27-s + 3.40·28-s − 2.22·29-s + 1.07·31-s − 1.06·32-s + 2.08·33-s − 2.05·34-s + 1.97·37-s − 1.94·38-s − 3.70·42-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.591686775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.591686775\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 16 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 103 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 96 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 80 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 132 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 295 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 56 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.935001634114670014301749078611, −7.80366694230644971267200281217, −7.48055651906683752018593183982, −7.22166928412123781669066765499, −6.80641808270507551526578347551, −6.47675308056408913045695037568, −5.77179584881438862264453518382, −5.67278254943997710947392829873, −5.22813989070521753434387550577, −4.88795858040285611245616278232, −4.34656374736070567874193758043, −4.00976060686471117412418566200, −3.37401466815977721440861306340, −3.31913045626660284087648110089, −2.59046403397414039205170155745, −2.48640336881630529349935287864, −1.77597565448545206966358290055, −1.34962702958601886699772344905, −1.05523977213570428593308749619, −0.873919183151157023377589516158,
0.873919183151157023377589516158, 1.05523977213570428593308749619, 1.34962702958601886699772344905, 1.77597565448545206966358290055, 2.48640336881630529349935287864, 2.59046403397414039205170155745, 3.31913045626660284087648110089, 3.37401466815977721440861306340, 4.00976060686471117412418566200, 4.34656374736070567874193758043, 4.88795858040285611245616278232, 5.22813989070521753434387550577, 5.67278254943997710947392829873, 5.77179584881438862264453518382, 6.47675308056408913045695037568, 6.80641808270507551526578347551, 7.22166928412123781669066765499, 7.48055651906683752018593183982, 7.80366694230644971267200281217, 7.935001634114670014301749078611
