| L(s) = 1 | − 2-s + 4-s − 2·5-s + 2·7-s − 3·8-s + 2·10-s + 7·11-s − 4·13-s − 2·14-s + 16-s + 6·17-s + 2·19-s − 2·20-s − 7·22-s − 9·23-s + 3·25-s + 4·26-s + 2·28-s − 2·29-s + 8·31-s + 32-s − 6·34-s − 4·35-s + 9·37-s − 2·38-s + 6·40-s − 9·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.755·7-s − 1.06·8-s + 0.632·10-s + 2.11·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.447·20-s − 1.49·22-s − 1.87·23-s + 3/5·25-s + 0.784·26-s + 0.377·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.676·35-s + 1.47·37-s − 0.324·38-s + 0.948·40-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.449464158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.449464158\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 64 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 132 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 128 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - T + 162 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 156 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−9.734799860980279499639096330989, −9.651664432277461380369749395776, −8.857210226163958058104389956260, −8.709696071474157382559872372758, −8.310415183645855532279745268019, −7.80880343512888857576556238814, −7.39822218725820446397430543655, −7.24687457357214400159316646837, −6.66057547523484338102983413680, −6.10136580954572760168751076281, −5.85732761548949942167758772339, −5.35104596423071181388411407641, −4.48827220794765230873410343403, −4.28577465182772932857620041190, −3.86362642456003291750199445001, −3.21328009972045559003412485900, −2.66775800460361348773672474845, −2.02181087587075838974186157154, −1.23560909985049987109707972196, −0.64203889538914671024816812879,
0.64203889538914671024816812879, 1.23560909985049987109707972196, 2.02181087587075838974186157154, 2.66775800460361348773672474845, 3.21328009972045559003412485900, 3.86362642456003291750199445001, 4.28577465182772932857620041190, 4.48827220794765230873410343403, 5.35104596423071181388411407641, 5.85732761548949942167758772339, 6.10136580954572760168751076281, 6.66057547523484338102983413680, 7.24687457357214400159316646837, 7.39822218725820446397430543655, 7.80880343512888857576556238814, 8.310415183645855532279745268019, 8.709696071474157382559872372758, 8.857210226163958058104389956260, 9.651664432277461380369749395776, 9.734799860980279499639096330989
