| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 2·6-s − 2·7-s − 4·8-s − 4·9-s + 5·11-s + 3·12-s + 8·13-s + 4·14-s + 5·16-s − 2·17-s + 8·18-s − 5·19-s − 2·21-s − 10·22-s + 23-s − 4·24-s − 16·26-s − 6·27-s − 6·28-s − 15·29-s − 3·31-s − 6·32-s + 5·33-s + 4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.816·6-s − 0.755·7-s − 1.41·8-s − 4/3·9-s + 1.50·11-s + 0.866·12-s + 2.21·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s + 1.88·18-s − 1.14·19-s − 0.436·21-s − 2.13·22-s + 0.208·23-s − 0.816·24-s − 3.13·26-s − 1.15·27-s − 1.13·28-s − 2.78·29-s − 0.538·31-s − 1.06·32-s + 0.870·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.434876675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.434876675\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 15 T + 113 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 63 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 127 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 23 T + 225 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 105 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 29 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 138 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 93 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 22 T + 310 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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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
−8.333575814124483585411931174224, −8.039549806855611025794068383354, −7.59692514564112764894154841272, −7.50415798371940459550210105103, −6.71943968898559988224871106506, −6.54660739177799727647604882167, −6.26470655964472919859832367753, −6.01980232875009895570728178713, −5.58675506825152689477044923671, −5.37773940245435635803738260501, −4.36331920553408062507659445431, −3.96653708075513386703010465378, −3.66643299815350019389291629711, −3.57507863597351081943703515736, −2.71782838675024804782172270138, −2.64787076906980121538658986970, −1.79623173231979591816333153055, −1.76114258179462970382122331069, −0.864827427309437020103090641384, −0.46709117940746626325711448733,
0.46709117940746626325711448733, 0.864827427309437020103090641384, 1.76114258179462970382122331069, 1.79623173231979591816333153055, 2.64787076906980121538658986970, 2.71782838675024804782172270138, 3.57507863597351081943703515736, 3.66643299815350019389291629711, 3.96653708075513386703010465378, 4.36331920553408062507659445431, 5.37773940245435635803738260501, 5.58675506825152689477044923671, 6.01980232875009895570728178713, 6.26470655964472919859832367753, 6.54660739177799727647604882167, 6.71943968898559988224871106506, 7.50415798371940459550210105103, 7.59692514564112764894154841272, 8.039549806855611025794068383354, 8.333575814124483585411931174224
