| L(s) = 1 | + 2-s + 3-s + 6-s + 5·7-s + 8-s − 2·9-s − 2·11-s + 10·13-s + 5·14-s − 16-s + 3·17-s − 2·18-s − 2·19-s + 5·21-s − 2·22-s − 11·23-s + 24-s + 10·26-s − 2·27-s − 9·29-s + 6·31-s − 6·32-s − 2·33-s + 3·34-s + 12·37-s − 2·38-s + 10·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.408·6-s + 1.88·7-s + 0.353·8-s − 2/3·9-s − 0.603·11-s + 2.77·13-s + 1.33·14-s − 1/4·16-s + 0.727·17-s − 0.471·18-s − 0.458·19-s + 1.09·21-s − 0.426·22-s − 2.29·23-s + 0.204·24-s + 1.96·26-s − 0.384·27-s − 1.67·29-s + 1.07·31-s − 1.06·32-s − 0.348·33-s + 0.514·34-s + 1.97·37-s − 0.324·38-s + 1.60·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.009258826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.009258826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 11 T + 73 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 9 T + 49 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 159 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 187 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 27 T + 373 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
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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
−11.78569107765125652198036962703, −11.75264289167998397713824240051, −11.24933123400379278318604389692, −10.83460400463024479003249245451, −10.48247638226480207491174066169, −9.828676751240870473559103235701, −8.872738272852148831121062500243, −8.843291587354579709770843927933, −8.092446463864149651351578020368, −7.78167051977930937994868351532, −7.76138672389568340003500471012, −6.36993791897891590419911293865, −6.02203397942302661482326578065, −5.62137348147238836438259296365, −4.80175039695978728299343645479, −4.36728592533815128393952288598, −3.79894828953799276588994147306, −3.17737129625561151037734333671, −2.04290806640930588735681357089, −1.51170863797042810712464977372,
1.51170863797042810712464977372, 2.04290806640930588735681357089, 3.17737129625561151037734333671, 3.79894828953799276588994147306, 4.36728592533815128393952288598, 4.80175039695978728299343645479, 5.62137348147238836438259296365, 6.02203397942302661482326578065, 6.36993791897891590419911293865, 7.76138672389568340003500471012, 7.78167051977930937994868351532, 8.092446463864149651351578020368, 8.843291587354579709770843927933, 8.872738272852148831121062500243, 9.828676751240870473559103235701, 10.48247638226480207491174066169, 10.83460400463024479003249245451, 11.24933123400379278318604389692, 11.75264289167998397713824240051, 11.78569107765125652198036962703
