| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 4·5-s + 2·6-s − 3·7-s + 4·8-s − 9-s + 8·10-s + 3·12-s + 3·13-s − 6·14-s + 4·15-s + 5·16-s − 3·17-s − 2·18-s + 2·19-s + 12·20-s − 3·21-s + 7·23-s + 4·24-s + 2·25-s + 6·26-s − 9·28-s − 29-s + 8·30-s + 4·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 1.78·5-s + 0.816·6-s − 1.13·7-s + 1.41·8-s − 1/3·9-s + 2.52·10-s + 0.866·12-s + 0.832·13-s − 1.60·14-s + 1.03·15-s + 5/4·16-s − 0.727·17-s − 0.471·18-s + 0.458·19-s + 2.68·20-s − 0.654·21-s + 1.45·23-s + 0.816·24-s + 2/5·25-s + 1.17·26-s − 1.70·28-s − 0.185·29-s + 1.46·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21141604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.37813122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.37813122\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 98 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 120 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 186 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 186 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.388013372165264433451553011106, −8.354379142627072384789793367177, −7.60694106066221753151363001908, −7.11753857845781992089867549780, −6.92851520915621082450923490537, −6.60792944412009385019188220043, −6.08598271570283301646587712261, −5.96360662406965274345005296603, −5.65306893796962804722356948928, −5.30865246335906277681277042760, −4.77555400944879620371328021404, −4.49550810085868735906681271510, −3.75247975615199999890362818538, −3.63032619239774462019841399777, −3.01544449142983945833167266384, −2.85846560900900947603954021878, −2.23629020885531582096547969005, −2.07223502052819286353448144161, −1.37839871735081611330517311973, −0.72945090124975956079557439977,
0.72945090124975956079557439977, 1.37839871735081611330517311973, 2.07223502052819286353448144161, 2.23629020885531582096547969005, 2.85846560900900947603954021878, 3.01544449142983945833167266384, 3.63032619239774462019841399777, 3.75247975615199999890362818538, 4.49550810085868735906681271510, 4.77555400944879620371328021404, 5.30865246335906277681277042760, 5.65306893796962804722356948928, 5.96360662406965274345005296603, 6.08598271570283301646587712261, 6.60792944412009385019188220043, 6.92851520915621082450923490537, 7.11753857845781992089867549780, 7.60694106066221753151363001908, 8.354379142627072384789793367177, 8.388013372165264433451553011106
