| L(s) = 1 | − 4·3-s − 2·5-s + 4·7-s + 8·9-s − 4·11-s + 4·13-s + 8·15-s − 4·17-s − 2·19-s − 16·21-s + 12·23-s + 3·25-s − 12·27-s − 4·29-s + 8·31-s + 16·33-s − 8·35-s + 12·37-s − 16·39-s − 4·41-s + 4·43-s − 16·45-s + 4·47-s + 6·49-s + 16·51-s − 4·53-s + 8·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.894·5-s + 1.51·7-s + 8/3·9-s − 1.20·11-s + 1.10·13-s + 2.06·15-s − 0.970·17-s − 0.458·19-s − 3.49·21-s + 2.50·23-s + 3/5·25-s − 2.30·27-s − 0.742·29-s + 1.43·31-s + 2.78·33-s − 1.35·35-s + 1.97·37-s − 2.56·39-s − 0.624·41-s + 0.609·43-s − 2.38·45-s + 0.583·47-s + 6/7·49-s + 2.24·51-s − 0.549·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.332209991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.332209991\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 154 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 136 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 212 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.975635625325628956910200148355, −7.947861460022356608331365400666, −7.56816008764371744985267929475, −7.02988009588609255489198835859, −6.73293544784854002754262875083, −6.57475369767368243362448794795, −5.93006165318933620392847525791, −5.76684419830009614012899265730, −5.34177035329069028368614833034, −4.98504766596448602259716081332, −4.77616891203080074289430267124, −4.53160436639570970899445619017, −3.96596839983123294524267839864, −3.75088571733376059673956504636, −2.87068078195394340863856827814, −2.59335884053265231702517697252, −1.93929606032299414233200589226, −1.28757276592574641033048930213, −0.66755555512537396191402850870, −0.61401219380394009344543808522,
0.61401219380394009344543808522, 0.66755555512537396191402850870, 1.28757276592574641033048930213, 1.93929606032299414233200589226, 2.59335884053265231702517697252, 2.87068078195394340863856827814, 3.75088571733376059673956504636, 3.96596839983123294524267839864, 4.53160436639570970899445619017, 4.77616891203080074289430267124, 4.98504766596448602259716081332, 5.34177035329069028368614833034, 5.76684419830009614012899265730, 5.93006165318933620392847525791, 6.57475369767368243362448794795, 6.73293544784854002754262875083, 7.02988009588609255489198835859, 7.56816008764371744985267929475, 7.947861460022356608331365400666, 7.975635625325628956910200148355
