| L(s) = 1 | + 4-s − 2·9-s + 2·19-s + 2·31-s − 2·36-s − 2·41-s + 49-s + 2·59-s − 64-s − 2·71-s + 2·76-s + 3·81-s − 2·101-s + 2·109-s + 2·121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s − 2·169-s + ⋯ |
| L(s) = 1 | + 4-s − 2·9-s + 2·19-s + 2·31-s − 2·36-s − 2·41-s + 49-s + 2·59-s − 64-s − 2·71-s + 2·76-s + 3·81-s − 2·101-s + 2·109-s + 2·121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s − 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.001009505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001009505\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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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
−10.88094925849513100475007253196, −10.31611005555746883682549417850, −9.745375666044587925445917159751, −9.736479655856421221788304309806, −8.708947130671630556292987312140, −8.695966476342903621559896603555, −8.301946708134444097654891517075, −7.64333590180011217867953441274, −7.28272962814183108869302836062, −6.82923687497072555163317355472, −6.36018236574266475286208299262, −5.80229200938498819358004024042, −5.60804298330032465500801713255, −4.99224872250018706175975538306, −4.52161811020678925869205945602, −3.47417145786663930246880135241, −3.22838924950130307453140274912, −2.64667848140516305275747409975, −2.21450966206841532962125146589, −1.12751217975685847919452715463,
1.12751217975685847919452715463, 2.21450966206841532962125146589, 2.64667848140516305275747409975, 3.22838924950130307453140274912, 3.47417145786663930246880135241, 4.52161811020678925869205945602, 4.99224872250018706175975538306, 5.60804298330032465500801713255, 5.80229200938498819358004024042, 6.36018236574266475286208299262, 6.82923687497072555163317355472, 7.28272962814183108869302836062, 7.64333590180011217867953441274, 8.301946708134444097654891517075, 8.695966476342903621559896603555, 8.708947130671630556292987312140, 9.736479655856421221788304309806, 9.745375666044587925445917159751, 10.31611005555746883682549417850, 10.88094925849513100475007253196
