L(s) = 1 | + 3·3-s − 5-s − 4·7-s + 6·9-s + 3·11-s + 4·13-s − 3·15-s + 6·17-s − 10·19-s − 12·21-s + 6·23-s + 9·27-s − 6·29-s + 2·31-s + 9·33-s + 4·35-s − 8·37-s + 12·39-s + 3·41-s + 11·43-s − 6·45-s + 7·49-s + 18·51-s + 12·53-s − 3·55-s − 30·57-s − 3·59-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.447·5-s − 1.51·7-s + 2·9-s + 0.904·11-s + 1.10·13-s − 0.774·15-s + 1.45·17-s − 2.29·19-s − 2.61·21-s + 1.25·23-s + 1.73·27-s − 1.11·29-s + 0.359·31-s + 1.56·33-s + 0.676·35-s − 1.31·37-s + 1.92·39-s + 0.468·41-s + 1.67·43-s − 0.894·45-s + 49-s + 2.52·51-s + 1.64·53-s − 0.404·55-s − 3.97·57-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.208526469\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.208526469\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.49897421605245869236952277247, −10.21289780206880567216753569045, −9.417541308764779902415699585209, −9.353629583556061453203500068952, −8.948955554398129765124944093313, −8.435858603904231763429045115396, −8.344613829901136435493235014713, −7.55870232114401753705904273887, −7.17977942168347120137183831319, −6.88314253294859671971811849541, −6.12212245074065086345496674465, −6.08023312401024454537753387710, −5.17998351658346428593068851196, −4.28873226265988229012981394407, −3.87818085717095173237037694826, −3.63661034085897527631193317907, −3.16098389543368325827477851598, −2.54007572311196241797055661688, −1.83632829658934941938625660347, −0.886081282075011703012248408510,
0.886081282075011703012248408510, 1.83632829658934941938625660347, 2.54007572311196241797055661688, 3.16098389543368325827477851598, 3.63661034085897527631193317907, 3.87818085717095173237037694826, 4.28873226265988229012981394407, 5.17998351658346428593068851196, 6.08023312401024454537753387710, 6.12212245074065086345496674465, 6.88314253294859671971811849541, 7.17977942168347120137183831319, 7.55870232114401753705904273887, 8.344613829901136435493235014713, 8.435858603904231763429045115396, 8.948955554398129765124944093313, 9.353629583556061453203500068952, 9.417541308764779902415699585209, 10.21289780206880567216753569045, 10.49897421605245869236952277247