| L(s) = 1 | + 9-s + 6·13-s + 2·17-s − 12·29-s − 10·37-s + 8·41-s + 10·49-s + 18·53-s + 12·61-s − 18·73-s + 81-s + 20·89-s + 6·97-s − 4·101-s + 4·109-s − 14·113-s + 6·117-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 1.66·13-s + 0.485·17-s − 2.22·29-s − 1.64·37-s + 1.24·41-s + 10/7·49-s + 2.47·53-s + 1.53·61-s − 2.10·73-s + 1/9·81-s + 2.11·89-s + 0.609·97-s − 0.398·101-s + 0.383·109-s − 1.31·113-s + 0.554·117-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.161·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.392915504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.392915504\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−7.83540487041894339635689883732, −7.41821810397637105155494485920, −7.19929019076115046584976214862, −6.66395544733240397854523548625, −6.10837574039395591018955960628, −5.66019988425805891741289162331, −5.50531724902698633695346568170, −4.88685525944060865041314880641, −4.06522363185768058926557610456, −3.83068347409164253221398360600, −3.55761953444439858212278594982, −2.72632634991445829166818625811, −2.07218279554419329920761833926, −1.46256558013774941495503677051, −0.71054314229633382874325179743,
0.71054314229633382874325179743, 1.46256558013774941495503677051, 2.07218279554419329920761833926, 2.72632634991445829166818625811, 3.55761953444439858212278594982, 3.83068347409164253221398360600, 4.06522363185768058926557610456, 4.88685525944060865041314880641, 5.50531724902698633695346568170, 5.66019988425805891741289162331, 6.10837574039395591018955960628, 6.66395544733240397854523548625, 7.19929019076115046584976214862, 7.41821810397637105155494485920, 7.83540487041894339635689883732
