L(s) = 1 | + 2·5-s − 3·7-s − 4·11-s − 4·13-s + 17-s + 6·19-s + 2·23-s + 3·25-s − 5·29-s + 5·31-s − 6·35-s − 5·37-s + 41-s + 4·47-s − 3·49-s + 11·53-s − 8·55-s − 59-s − 12·61-s − 8·65-s + 11·67-s + 71-s − 20·73-s + 12·77-s + 2·79-s − 83-s + 2·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.13·7-s − 1.20·11-s − 1.10·13-s + 0.242·17-s + 1.37·19-s + 0.417·23-s + 3/5·25-s − 0.928·29-s + 0.898·31-s − 1.01·35-s − 0.821·37-s + 0.156·41-s + 0.583·47-s − 3/7·49-s + 1.51·53-s − 1.07·55-s − 0.130·59-s − 1.53·61-s − 0.992·65-s + 1.34·67-s + 0.118·71-s − 2.34·73-s + 1.36·77-s + 0.225·79-s − 0.109·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 42 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 44 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 132 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 114 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 11 T + 126 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T + 128 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.55132580635275954862075241235, −7.25218088851627039901618615753, −6.86848693798995655435774058989, −6.82215457643371986235252294989, −6.06819817707968183609378659976, −5.97438212501618380880049706157, −5.43035571150189117676523009541, −5.38946408519859782195985768422, −4.78097433207281832785876348386, −4.78075561090295124072155898060, −3.88785124007004315336869909067, −3.74997968826060958594501578097, −2.99239150551935389888065931260, −2.95598604514525345932257231851, −2.48154195133559453217969924620, −2.23627752532550617076991639109, −1.34206603356318171619370297079, −1.16720060058304513738909068305, 0, 0,
1.16720060058304513738909068305, 1.34206603356318171619370297079, 2.23627752532550617076991639109, 2.48154195133559453217969924620, 2.95598604514525345932257231851, 2.99239150551935389888065931260, 3.74997968826060958594501578097, 3.88785124007004315336869909067, 4.78075561090295124072155898060, 4.78097433207281832785876348386, 5.38946408519859782195985768422, 5.43035571150189117676523009541, 5.97438212501618380880049706157, 6.06819817707968183609378659976, 6.82215457643371986235252294989, 6.86848693798995655435774058989, 7.25218088851627039901618615753, 7.55132580635275954862075241235