| L(s) = 1 | + 2·5-s − 2·7-s − 2·11-s − 6·13-s + 4·17-s − 6·23-s + 3·25-s + 10·29-s + 4·31-s − 4·35-s − 4·37-s + 2·41-s − 6·43-s − 4·47-s + 2·49-s − 4·53-s − 4·55-s + 10·59-s − 6·61-s − 12·65-s − 16·67-s + 22·71-s + 18·73-s + 4·77-s + 16·79-s − 6·83-s + 8·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.603·11-s − 1.66·13-s + 0.970·17-s − 1.25·23-s + 3/5·25-s + 1.85·29-s + 0.718·31-s − 0.676·35-s − 0.657·37-s + 0.312·41-s − 0.914·43-s − 0.583·47-s + 2/7·49-s − 0.549·53-s − 0.539·55-s + 1.30·59-s − 0.768·61-s − 1.48·65-s − 1.95·67-s + 2.61·71-s + 2.10·73-s + 0.455·77-s + 1.80·79-s − 0.658·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74649600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74649600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.175117689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.175117689\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 53 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 97 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 209 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.84847845101030319970855690256, −7.72127832209195856679055978387, −7.23859352116110682988485746200, −6.85599532133692295764126379370, −6.41234017315267260237338049642, −6.35776570370488682480872670930, −5.97237679585719654625152095206, −5.49707847128187198974862520825, −5.07236283868000528192444173609, −4.90861076107594825783883715405, −4.65677851425858007191278466393, −4.07759199465766424727716474186, −3.41273457693379824134297973430, −3.33253155000014460833495803080, −2.81920074450533144827474709722, −2.41684489052065841372334930484, −2.02238362532470220689041449115, −1.73163017943011114188459553787, −0.64059663797803248254136997257, −0.63493935818306436098045623596,
0.63493935818306436098045623596, 0.64059663797803248254136997257, 1.73163017943011114188459553787, 2.02238362532470220689041449115, 2.41684489052065841372334930484, 2.81920074450533144827474709722, 3.33253155000014460833495803080, 3.41273457693379824134297973430, 4.07759199465766424727716474186, 4.65677851425858007191278466393, 4.90861076107594825783883715405, 5.07236283868000528192444173609, 5.49707847128187198974862520825, 5.97237679585719654625152095206, 6.35776570370488682480872670930, 6.41234017315267260237338049642, 6.85599532133692295764126379370, 7.23859352116110682988485746200, 7.72127832209195856679055978387, 7.84847845101030319970855690256
