| L(s) = 1 | − 2-s + 5-s − 7-s + 8-s − 10-s − 3·11-s + 14-s − 16-s + 3·22-s − 3·31-s − 35-s + 40-s − 53-s − 3·55-s − 56-s + 59-s + 3·62-s + 64-s + 70-s + 2·73-s + 3·77-s − 79-s − 80-s − 3·88-s − 2·97-s − 2·101-s + 2·103-s + ⋯ |
| L(s) = 1 | − 2-s + 5-s − 7-s + 8-s − 10-s − 3·11-s + 14-s − 16-s + 3·22-s − 3·31-s − 35-s + 40-s − 53-s − 3·55-s − 56-s + 59-s + 3·62-s + 64-s + 70-s + 2·73-s + 3·77-s − 79-s − 80-s − 3·88-s − 2·97-s − 2·101-s + 2·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2495996073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2495996073\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−9.632920543496502484663595866130, −8.941085060753119121150840365324, −8.606402666192640700344768862547, −8.120698976982002192957729464377, −7.71422151452433866373053179238, −7.67685506422168550048407575987, −7.00142651918653986204386683242, −6.83640181730218129581832489682, −6.19188885543868243695005092675, −5.58577881417783905808875067267, −5.53744930580630495810310083325, −5.10914910299549229702469384201, −4.77909387233683719618008975357, −3.93035864884599878931306877547, −3.64967509166395519374445727408, −2.83320836993812420942846545150, −2.65723026404817183797940640211, −1.98719967937186098026503117062, −1.63141028656279175230985336625, −0.37090590848857770032086575914,
0.37090590848857770032086575914, 1.63141028656279175230985336625, 1.98719967937186098026503117062, 2.65723026404817183797940640211, 2.83320836993812420942846545150, 3.64967509166395519374445727408, 3.93035864884599878931306877547, 4.77909387233683719618008975357, 5.10914910299549229702469384201, 5.53744930580630495810310083325, 5.58577881417783905808875067267, 6.19188885543868243695005092675, 6.83640181730218129581832489682, 7.00142651918653986204386683242, 7.67685506422168550048407575987, 7.71422151452433866373053179238, 8.120698976982002192957729464377, 8.606402666192640700344768862547, 8.941085060753119121150840365324, 9.632920543496502484663595866130
