| L(s) = 1 | − 2-s + 2·3-s − 2·6-s − 8-s + 3·9-s − 6·11-s + 2·13-s − 16-s + 2·17-s − 3·18-s − 6·19-s + 6·22-s − 2·23-s − 2·24-s − 2·26-s + 4·27-s − 8·29-s + 6·32-s − 12·33-s − 2·34-s − 6·37-s + 6·38-s + 4·39-s − 12·43-s + 2·46-s + 14·47-s − 2·48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 0.816·6-s − 0.353·8-s + 9-s − 1.80·11-s + 0.554·13-s − 1/4·16-s + 0.485·17-s − 0.707·18-s − 1.37·19-s + 1.27·22-s − 0.417·23-s − 0.408·24-s − 0.392·26-s + 0.769·27-s − 1.48·29-s + 1.06·32-s − 2.08·33-s − 0.342·34-s − 0.986·37-s + 0.973·38-s + 0.640·39-s − 1.82·43-s + 0.294·46-s + 2.04·47-s − 0.288·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 24 T + 265 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 206 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.378411721943328876613174304594, −8.275578923869579581678649967951, −7.52492988208389317279785537834, −7.44613662530466639315094134783, −7.22325177741589594202884447016, −6.48925868697257161266471191489, −6.23443713399125124615641648240, −5.74995990895694318809327625707, −5.21000062545464898493878138059, −5.13421765817438793379118565886, −4.24915640129946265134040950785, −4.08804768540649445920529278707, −3.68896079250887193015175359670, −2.95401565305860381712198420514, −2.75450300721652680578859920863, −2.34289351708495329933987444230, −1.76274702040264180908585726671, −1.28348880225154921514410835911, 0, 0,
1.28348880225154921514410835911, 1.76274702040264180908585726671, 2.34289351708495329933987444230, 2.75450300721652680578859920863, 2.95401565305860381712198420514, 3.68896079250887193015175359670, 4.08804768540649445920529278707, 4.24915640129946265134040950785, 5.13421765817438793379118565886, 5.21000062545464898493878138059, 5.74995990895694318809327625707, 6.23443713399125124615641648240, 6.48925868697257161266471191489, 7.22325177741589594202884447016, 7.44613662530466639315094134783, 7.52492988208389317279785537834, 8.275578923869579581678649967951, 8.378411721943328876613174304594
