| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 4·11-s + 6·13-s + 5·16-s + 6·17-s + 12·19-s − 8·22-s − 2·23-s − 8·25-s + 12·26-s − 2·29-s + 6·31-s + 6·32-s + 12·34-s − 4·37-s + 24·38-s − 2·43-s − 12·44-s − 4·46-s − 16·50-s + 18·52-s + 14·53-s − 4·58-s + 6·59-s + 12·61-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1.20·11-s + 1.66·13-s + 5/4·16-s + 1.45·17-s + 2.75·19-s − 1.70·22-s − 0.417·23-s − 8/5·25-s + 2.35·26-s − 0.371·29-s + 1.07·31-s + 1.06·32-s + 2.05·34-s − 0.657·37-s + 3.89·38-s − 0.304·43-s − 1.80·44-s − 0.589·46-s − 2.26·50-s + 2.49·52-s + 1.92·53-s − 0.525·58-s + 0.781·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.022405760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.022405760\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 119 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 209 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 330 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−8.904402627740010420481192853669, −8.622644129773862539529832504706, −8.012897113797102669409786590718, −7.79379376956131483533436271788, −7.59407855485482138496048055403, −7.17410808353620848711608687012, −6.43660470973841071689009844649, −6.43294557206771289998648121994, −5.58889081562060924951492103563, −5.58091837216949866736706019275, −5.21895273110017556064096732903, −5.02328257322575875980958178394, −4.05524389291606364540711530746, −3.82454224164207829957955491269, −3.45810404312383599362920006109, −3.19336032846081947983380085263, −2.46674086760496643756866727522, −2.12787860985199687341077284804, −1.20995913547507016240539929411, −0.890384696189987934251088519419,
0.890384696189987934251088519419, 1.20995913547507016240539929411, 2.12787860985199687341077284804, 2.46674086760496643756866727522, 3.19336032846081947983380085263, 3.45810404312383599362920006109, 3.82454224164207829957955491269, 4.05524389291606364540711530746, 5.02328257322575875980958178394, 5.21895273110017556064096732903, 5.58091837216949866736706019275, 5.58889081562060924951492103563, 6.43294557206771289998648121994, 6.43660470973841071689009844649, 7.17410808353620848711608687012, 7.59407855485482138496048055403, 7.79379376956131483533436271788, 8.012897113797102669409786590718, 8.622644129773862539529832504706, 8.904402627740010420481192853669
