| L(s) = 1 | − 5-s + 2·7-s + 9-s − 2·11-s − 19-s + 25-s − 2·35-s + 2·43-s − 45-s + 49-s − 3·53-s + 2·55-s + 2·63-s − 4·77-s − 3·79-s + 2·83-s + 95-s + 3·97-s − 2·99-s + 3·121-s − 2·125-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 5-s + 2·7-s + 9-s − 2·11-s − 19-s + 25-s − 2·35-s + 2·43-s − 45-s + 49-s − 3·53-s + 2·55-s + 2·63-s − 4·77-s − 3·79-s + 2·83-s + 95-s + 3·97-s − 2·99-s + 3·121-s − 2·125-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.246777853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.246777853\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−8.794732558489193085887053209647, −8.557382882040194823215745021452, −7.959388314512843262717855707739, −7.911448115160538493617567579201, −7.71451313757838681602529753278, −7.32229208299773081309552172946, −6.96093219636892474800478845206, −6.35300844636509689178891268290, −5.97948721888090944897322787390, −5.41885627377976141548132539303, −5.06481516689991472056421212404, −4.63180437365456102900140970278, −4.49243754141138424659596183661, −4.22541799328451635045985275635, −3.47166690818274085829073388825, −3.02058738725274995818374435469, −2.50489588765767424931020944921, −1.86901931898415632542501174022, −1.62089124365510548962845749433, −0.67650917319613382229966624947,
0.67650917319613382229966624947, 1.62089124365510548962845749433, 1.86901931898415632542501174022, 2.50489588765767424931020944921, 3.02058738725274995818374435469, 3.47166690818274085829073388825, 4.22541799328451635045985275635, 4.49243754141138424659596183661, 4.63180437365456102900140970278, 5.06481516689991472056421212404, 5.41885627377976141548132539303, 5.97948721888090944897322787390, 6.35300844636509689178891268290, 6.96093219636892474800478845206, 7.32229208299773081309552172946, 7.71451313757838681602529753278, 7.911448115160538493617567579201, 7.959388314512843262717855707739, 8.557382882040194823215745021452, 8.794732558489193085887053209647
