| L(s) = 1 | + 3-s + 7-s − 9-s − 4·11-s + 3·13-s − 17-s − 12·19-s + 21-s + 2·23-s + 4·29-s + 2·31-s − 4·33-s + 5·37-s + 3·39-s − 3·47-s − 9·49-s − 51-s + 5·53-s − 12·57-s − 5·59-s − 6·61-s − 63-s − 13·67-s + 2·69-s − 12·71-s + 13·73-s − 4·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 1/3·9-s − 1.20·11-s + 0.832·13-s − 0.242·17-s − 2.75·19-s + 0.218·21-s + 0.417·23-s + 0.742·29-s + 0.359·31-s − 0.696·33-s + 0.821·37-s + 0.480·39-s − 0.437·47-s − 9/7·49-s − 0.140·51-s + 0.686·53-s − 1.58·57-s − 0.650·59-s − 0.768·61-s − 0.125·63-s − 1.58·67-s + 0.240·69-s − 1.42·71-s + 1.52·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 86 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 13 T + 172 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 161 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 13 T + 184 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−7.67515674178765119626732450532, −7.34065903525012664241206510011, −6.70517588459698362817480769326, −6.52873853650608910052064002523, −6.25099667019634192060068642299, −5.96571217505450556679785822764, −5.40153527856427631922974128368, −5.18276117023756711582933075599, −4.60238915503192464739791373651, −4.41956917631223182757267956738, −4.14876561503931504709375167783, −3.64940498026685793510273396688, −2.95705666258258593086634188239, −2.95080618722928982380607222207, −2.43733381327145779335448920140, −2.10789318919179852705781179841, −1.51094332684059179443101783081, −1.12075480713213196638785670043, 0, 0,
1.12075480713213196638785670043, 1.51094332684059179443101783081, 2.10789318919179852705781179841, 2.43733381327145779335448920140, 2.95080618722928982380607222207, 2.95705666258258593086634188239, 3.64940498026685793510273396688, 4.14876561503931504709375167783, 4.41956917631223182757267956738, 4.60238915503192464739791373651, 5.18276117023756711582933075599, 5.40153527856427631922974128368, 5.96571217505450556679785822764, 6.25099667019634192060068642299, 6.52873853650608910052064002523, 6.70517588459698362817480769326, 7.34065903525012664241206510011, 7.67515674178765119626732450532
