| L(s) = 1 | − 4·13-s − 16-s + 4·37-s − 4·73-s + 4·97-s − 8·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4·13-s − 16-s + 4·37-s − 4·73-s + 4·97-s − 8·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 4·208-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6457699447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6457699447\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.50294492265515532921569527455, −6.42079015610059934453004134586, −6.26221289577841511216967252075, −5.86849156404465529247899925880, −5.76111682455943732734931546728, −5.50725697215815301442372474380, −5.42644942759043730688563123964, −4.90257690332917788093201432163, −4.76861709937762468948906299731, −4.73920655161142613111988552854, −4.68478505993463051904074843986, −4.33803645540880965960298904238, −4.00582112486346243382547084672, −3.90868668207991451019570444415, −3.74486610465797861744166082363, −2.98185927213182464138855572579, −2.83120276068323970914755691480, −2.79356229032005110392469651048, −2.77180630588933270405723835382, −2.30513384733070205137195182861, −2.14592049679446770969536672138, −1.76498917477933447539020768607, −1.51137226215006181899776792291, −0.907488786209727316730007950170, −0.41492668483403096388958338654,
0.41492668483403096388958338654, 0.907488786209727316730007950170, 1.51137226215006181899776792291, 1.76498917477933447539020768607, 2.14592049679446770969536672138, 2.30513384733070205137195182861, 2.77180630588933270405723835382, 2.79356229032005110392469651048, 2.83120276068323970914755691480, 2.98185927213182464138855572579, 3.74486610465797861744166082363, 3.90868668207991451019570444415, 4.00582112486346243382547084672, 4.33803645540880965960298904238, 4.68478505993463051904074843986, 4.73920655161142613111988552854, 4.76861709937762468948906299731, 4.90257690332917788093201432163, 5.42644942759043730688563123964, 5.50725697215815301442372474380, 5.76111682455943732734931546728, 5.86849156404465529247899925880, 6.26221289577841511216967252075, 6.42079015610059934453004134586, 6.50294492265515532921569527455
