| L(s) = 1 | − 2-s − 5-s + 8-s + 10-s − 4·11-s + 6·13-s − 16-s + 3·17-s − 2·19-s + 4·22-s + 12·23-s + 5·25-s − 6·26-s + 18·29-s + 20·31-s − 3·34-s − 37-s + 2·38-s − 40-s + 3·41-s − 16·43-s − 12·46-s − 4·47-s + 7·49-s − 5·50-s − 6·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.447·5-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.66·13-s − 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.852·22-s + 2.50·23-s + 25-s − 1.17·26-s + 3.34·29-s + 3.59·31-s − 0.514·34-s − 0.164·37-s + 0.324·38-s − 0.158·40-s + 0.468·41-s − 2.43·43-s − 1.76·46-s − 0.583·47-s + 49-s − 0.707·50-s − 0.824·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.423899452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.423899452\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
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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
−10.56182993229973597377111061432, −10.43137175751561101728152793698, −9.842415721938944643242407657148, −9.620051478534108353908660449138, −8.605848493459808768427459796294, −8.517268320953162470790712985072, −8.250448953941239666876287998488, −8.142501523555363276404054290578, −7.05122773389641291060436914852, −6.90405443356493407183197926199, −6.44391746946068000561569181949, −5.92060136837988213742049818426, −5.03788140805041894714386836674, −4.83800202762840085182671713931, −4.43072838136009752251176800975, −3.48482059807355046542627915642, −2.82793741257533133459222477835, −2.76348563949483340092135235285, −1.05977880508536725700771349528, −1.01936612968966133430867180724,
1.01936612968966133430867180724, 1.05977880508536725700771349528, 2.76348563949483340092135235285, 2.82793741257533133459222477835, 3.48482059807355046542627915642, 4.43072838136009752251176800975, 4.83800202762840085182671713931, 5.03788140805041894714386836674, 5.92060136837988213742049818426, 6.44391746946068000561569181949, 6.90405443356493407183197926199, 7.05122773389641291060436914852, 8.142501523555363276404054290578, 8.250448953941239666876287998488, 8.517268320953162470790712985072, 8.605848493459808768427459796294, 9.620051478534108353908660449138, 9.842415721938944643242407657148, 10.43137175751561101728152793698, 10.56182993229973597377111061432
