| L(s) = 1 | + 2·5-s + 6·7-s + 8·11-s − 4·13-s − 2·17-s − 2·19-s + 3·25-s − 4·29-s + 6·31-s + 12·35-s − 2·37-s − 4·41-s + 10·43-s + 14·47-s + 16·49-s + 10·53-s + 16·55-s + 20·59-s + 8·61-s − 8·65-s + 4·71-s + 2·73-s + 48·77-s − 24·79-s + 6·83-s − 4·85-s − 24·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2.26·7-s + 2.41·11-s − 1.10·13-s − 0.485·17-s − 0.458·19-s + 3/5·25-s − 0.742·29-s + 1.07·31-s + 2.02·35-s − 0.328·37-s − 0.624·41-s + 1.52·43-s + 2.04·47-s + 16/7·49-s + 1.37·53-s + 2.15·55-s + 2.60·59-s + 1.02·61-s − 0.992·65-s + 0.474·71-s + 0.234·73-s + 5.47·77-s − 2.70·79-s + 0.658·83-s − 0.433·85-s − 2.51·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.566769495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.566769495\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 140 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 128 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 20 T + 215 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 143 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 120 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.257732955305661071194815901146, −7.83212548830848661788801858763, −7.41454050442794743012981094168, −7.15263532728323849872830116358, −6.69803304949064143157976791002, −6.65404195137658875818003257071, −5.89940897309354659045703174434, −5.78369491055284435248755022553, −5.23927834504333523603928498456, −5.09011276534072940114270134798, −4.42858641733139790131815979186, −4.37394895455854117977047305371, −3.87922375646003223007308422966, −3.69018113547460033703331677457, −2.56199138821887536200673288295, −2.53851181793399744860707672988, −1.94141679084746957151344141938, −1.69390198017959022774483475940, −1.07503316483974931512592461512, −0.77903632400619241696552040506,
0.77903632400619241696552040506, 1.07503316483974931512592461512, 1.69390198017959022774483475940, 1.94141679084746957151344141938, 2.53851181793399744860707672988, 2.56199138821887536200673288295, 3.69018113547460033703331677457, 3.87922375646003223007308422966, 4.37394895455854117977047305371, 4.42858641733139790131815979186, 5.09011276534072940114270134798, 5.23927834504333523603928498456, 5.78369491055284435248755022553, 5.89940897309354659045703174434, 6.65404195137658875818003257071, 6.69803304949064143157976791002, 7.15263532728323849872830116358, 7.41454050442794743012981094168, 7.83212548830848661788801858763, 8.257732955305661071194815901146
