| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 3·5-s + 4·6-s + 3·7-s − 4·8-s + 3·9-s + 6·10-s − 4·11-s − 6·12-s − 2·13-s − 6·14-s + 6·15-s + 5·16-s − 9·17-s − 6·18-s + 19-s − 9·20-s − 6·21-s + 8·22-s − 2·23-s + 8·24-s + 25-s + 4·26-s − 4·27-s + 9·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.34·5-s + 1.63·6-s + 1.13·7-s − 1.41·8-s + 9-s + 1.89·10-s − 1.20·11-s − 1.73·12-s − 0.554·13-s − 1.60·14-s + 1.54·15-s + 5/4·16-s − 2.18·17-s − 1.41·18-s + 0.229·19-s − 2.01·20-s − 1.30·21-s + 1.70·22-s − 0.417·23-s + 1.63·24-s + 1/5·25-s + 0.784·26-s − 0.769·27-s + 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16016004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16016004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3730472380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3730472380\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - T + 70 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - T + 114 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 106 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 154 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 198 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.567679896474643777049040896674, −8.303341589837278045060842141891, −7.67398400599443689656561522125, −7.65792385984152722402758558428, −7.37105242710467648765308728155, −7.06766181261656614305781480476, −6.49462963917441556074533770339, −6.13572466273971234404264805809, −5.82759956286418490819774086309, −5.25524844064432678307146103636, −4.73343355001403294946793321509, −4.72563430170498713625600803696, −4.04359928656030092853794585067, −3.87625869951588457828931646757, −2.80035019239922803942416440636, −2.71880214285323271311588368928, −1.82761372203279003886868453172, −1.77875576855483576732294434416, −0.60888474720614785080771244918, −0.41768954893914248499916994782,
0.41768954893914248499916994782, 0.60888474720614785080771244918, 1.77875576855483576732294434416, 1.82761372203279003886868453172, 2.71880214285323271311588368928, 2.80035019239922803942416440636, 3.87625869951588457828931646757, 4.04359928656030092853794585067, 4.72563430170498713625600803696, 4.73343355001403294946793321509, 5.25524844064432678307146103636, 5.82759956286418490819774086309, 6.13572466273971234404264805809, 6.49462963917441556074533770339, 7.06766181261656614305781480476, 7.37105242710467648765308728155, 7.65792385984152722402758558428, 7.67398400599443689656561522125, 8.303341589837278045060842141891, 8.567679896474643777049040896674
