| L(s) = 1 | − 2·3-s − 2·4-s − 4·5-s + 6·7-s + 2·9-s + 4·12-s + 8·15-s + 3·16-s + 20·17-s + 8·20-s − 12·21-s − 6·27-s − 12·28-s − 24·35-s − 4·36-s + 4·41-s + 8·43-s − 8·45-s + 24·47-s − 6·48-s + 18·49-s − 40·51-s + 4·59-s − 16·60-s + 12·63-s − 4·64-s + 24·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 1.78·5-s + 2.26·7-s + 2/3·9-s + 1.15·12-s + 2.06·15-s + 3/4·16-s + 4.85·17-s + 1.78·20-s − 2.61·21-s − 1.15·27-s − 2.26·28-s − 4.05·35-s − 2/3·36-s + 0.624·41-s + 1.21·43-s − 1.19·45-s + 3.50·47-s − 0.866·48-s + 18/7·49-s − 5.60·51-s + 0.520·59-s − 2.06·60-s + 1.51·63-s − 1/2·64-s + 2.93·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.474441103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.474441103\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 - 44 T^{2} + 886 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 9038 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_4\times C_2$ | \( 1 - 76 T^{2} + 3766 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 16 T^{2} - 2354 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 6302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 89 | $D_{4}$ | \( ( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 340 T^{2} + 47398 T^{4} - 340 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.960195100297008361430514007303, −7.946442717532163595655263088628, −7.50927110551669740277263008710, −7.45913893790855839235613836933, −7.36203227983659590319131766139, −6.77201429867421611585224717694, −6.64786886128596789253957281411, −5.85051485121645107465174064908, −5.82434976170426261929661264938, −5.63996305178035765135593991040, −5.58059461009239174623110925924, −5.12833905337014648757307953939, −5.11231936387832528042773213988, −4.67859075981237580197284009778, −4.41055849278307176460139108080, −3.97153044460235755914528851188, −3.80645174564738535204300583265, −3.79730004063956671084240800959, −3.43125204184876553210421189297, −2.96460290260980045292561963437, −2.31279278911285762268133530575, −2.02024850973126980547445189196, −1.17292470199172224556447731944, −1.01607175494390905174026577253, −0.69097073518968889309072920179,
0.69097073518968889309072920179, 1.01607175494390905174026577253, 1.17292470199172224556447731944, 2.02024850973126980547445189196, 2.31279278911285762268133530575, 2.96460290260980045292561963437, 3.43125204184876553210421189297, 3.79730004063956671084240800959, 3.80645174564738535204300583265, 3.97153044460235755914528851188, 4.41055849278307176460139108080, 4.67859075981237580197284009778, 5.11231936387832528042773213988, 5.12833905337014648757307953939, 5.58059461009239174623110925924, 5.63996305178035765135593991040, 5.82434976170426261929661264938, 5.85051485121645107465174064908, 6.64786886128596789253957281411, 6.77201429867421611585224717694, 7.36203227983659590319131766139, 7.45913893790855839235613836933, 7.50927110551669740277263008710, 7.946442717532163595655263088628, 7.960195100297008361430514007303
