| L(s) = 1 | − 2-s − 4·5-s + 8-s + 4·10-s − 2·11-s + 6·13-s − 16-s − 5·17-s + 7·19-s + 2·22-s − 8·23-s + 2·25-s − 6·26-s − 4·29-s + 6·31-s + 5·34-s − 2·37-s − 7·38-s − 4·40-s + 3·41-s + 43-s + 8·46-s − 2·50-s + 12·53-s + 8·55-s + 4·58-s − 7·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.78·5-s + 0.353·8-s + 1.26·10-s − 0.603·11-s + 1.66·13-s − 1/4·16-s − 1.21·17-s + 1.60·19-s + 0.426·22-s − 1.66·23-s + 2/5·25-s − 1.17·26-s − 0.742·29-s + 1.07·31-s + 0.857·34-s − 0.328·37-s − 1.13·38-s − 0.632·40-s + 0.468·41-s + 0.152·43-s + 1.17·46-s − 0.282·50-s + 1.64·53-s + 1.07·55-s + 0.525·58-s − 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8673248323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8673248323\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 16 T + 173 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 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
−9.051773431572888299287554977646, −8.409856516810259143372194725757, −8.254029065573014026360465971436, −8.074185992594133302064861435516, −7.64026422292377683334903142510, −7.37361004215331648845711461191, −6.91278515068091232076092668255, −6.49628179168409664082774137112, −5.85529277817401938098218486611, −5.77643839851255798247956702338, −5.10993824603754659375049173356, −4.56624098498399379404468894281, −4.14951385249271210161872419794, −3.89810385030482013172080098630, −3.44865069292980736599652054521, −3.09940198233352821221106510789, −2.18582867691027681027792759570, −1.86448457309065429023780747652, −0.838039241967278125687629409221, −0.48561583303596323392159799103,
0.48561583303596323392159799103, 0.838039241967278125687629409221, 1.86448457309065429023780747652, 2.18582867691027681027792759570, 3.09940198233352821221106510789, 3.44865069292980736599652054521, 3.89810385030482013172080098630, 4.14951385249271210161872419794, 4.56624098498399379404468894281, 5.10993824603754659375049173356, 5.77643839851255798247956702338, 5.85529277817401938098218486611, 6.49628179168409664082774137112, 6.91278515068091232076092668255, 7.37361004215331648845711461191, 7.64026422292377683334903142510, 8.074185992594133302064861435516, 8.254029065573014026360465971436, 8.409856516810259143372194725757, 9.051773431572888299287554977646
