| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s − 3·11-s + 5·13-s − 16-s − 6·17-s + 6·19-s + 3·22-s + 5·23-s − 24-s − 5·26-s + 27-s − 2·29-s − 12·31-s + 3·33-s + 6·34-s + 9·37-s − 6·38-s − 5·39-s + 10·41-s + 8·43-s − 5·46-s − 24·47-s + 48-s + 7·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s + 0.353·8-s − 0.904·11-s + 1.38·13-s − 1/4·16-s − 1.45·17-s + 1.37·19-s + 0.639·22-s + 1.04·23-s − 0.204·24-s − 0.980·26-s + 0.192·27-s − 0.371·29-s − 2.15·31-s + 0.522·33-s + 1.02·34-s + 1.47·37-s − 0.973·38-s − 0.800·39-s + 1.56·41-s + 1.21·43-s − 0.737·46-s − 3.50·47-s + 0.144·48-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.283316557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283316557\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 7 T - 22 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.222777725480412276431749004986, −9.068019926046568275785908205087, −8.551128396813162501591837452109, −8.455681352980129562363016003516, −7.67628538647037566870581161856, −7.55062776956692897920057237579, −7.07352497070938636083969297020, −6.78468228197002332414592231305, −6.01893988474354640917960065732, −5.94687556300535898333615649533, −5.37003638082683059643936598143, −5.07617733788036352074993339007, −4.54225963208543052702061089057, −4.01070795575742340885248809369, −3.55583483336812145040453958499, −3.07135764370361871831691683793, −2.27666197333283320360815385626, −1.98005565414589211947868547933, −0.904470594180039076786691634809, −0.65640516450919566207764093842,
0.65640516450919566207764093842, 0.904470594180039076786691634809, 1.98005565414589211947868547933, 2.27666197333283320360815385626, 3.07135764370361871831691683793, 3.55583483336812145040453958499, 4.01070795575742340885248809369, 4.54225963208543052702061089057, 5.07617733788036352074993339007, 5.37003638082683059643936598143, 5.94687556300535898333615649533, 6.01893988474354640917960065732, 6.78468228197002332414592231305, 7.07352497070938636083969297020, 7.55062776956692897920057237579, 7.67628538647037566870581161856, 8.455681352980129562363016003516, 8.551128396813162501591837452109, 9.068019926046568275785908205087, 9.222777725480412276431749004986
