| L(s) = 1 | − 3-s + 5-s + 2·7-s − 4·9-s − 6·11-s + 2·13-s − 15-s + 2·17-s − 8·19-s − 2·21-s + 2·23-s − 8·25-s + 6·27-s − 2·29-s + 3·31-s + 6·33-s + 2·35-s + 8·37-s − 2·39-s + 11·41-s − 13·43-s − 4·45-s + 8·47-s + 3·49-s − 2·51-s + 3·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s − 4/3·9-s − 1.80·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 1.83·19-s − 0.436·21-s + 0.417·23-s − 8/5·25-s + 1.15·27-s − 0.371·29-s + 0.538·31-s + 1.04·33-s + 0.338·35-s + 1.31·37-s − 0.320·39-s + 1.71·41-s − 1.98·43-s − 0.596·45-s + 1.16·47-s + 3/7·49-s − 0.280·51-s + 0.412·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.705981118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.705981118\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11 T + 81 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T - 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 159 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 21 T + 245 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 19 T + 253 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.974224657931626322975793193405, −7.85321447766400466279884035026, −7.45952258110471161977219170812, −7.07108608951389518664019393260, −6.43328215411556502267113282878, −6.10743365925654954095622486285, −5.91475206560562586175191756142, −5.80776637951241252556581362967, −5.23995929346021743976510940407, −4.98042881469667387876819810878, −4.61045239395297084914147401505, −4.30136791525254213536169601384, −3.59647017718094309831269773891, −3.43135527213051253590442640954, −2.66819410727380007603645683899, −2.49338630976254685347299399477, −2.09183037419661615297929887122, −1.69010823860617216346377122733, −0.71719957243397953576759096319, −0.44895121116020088799937542157,
0.44895121116020088799937542157, 0.71719957243397953576759096319, 1.69010823860617216346377122733, 2.09183037419661615297929887122, 2.49338630976254685347299399477, 2.66819410727380007603645683899, 3.43135527213051253590442640954, 3.59647017718094309831269773891, 4.30136791525254213536169601384, 4.61045239395297084914147401505, 4.98042881469667387876819810878, 5.23995929346021743976510940407, 5.80776637951241252556581362967, 5.91475206560562586175191756142, 6.10743365925654954095622486285, 6.43328215411556502267113282878, 7.07108608951389518664019393260, 7.45952258110471161977219170812, 7.85321447766400466279884035026, 7.974224657931626322975793193405
