| L(s) = 1 | − 3-s − 4-s + 6·7-s + 9-s + 12-s + 4·13-s + 16-s + 4·19-s − 6·21-s − 2·25-s − 27-s − 6·28-s − 4·31-s − 36-s + 2·37-s − 4·39-s + 9·43-s − 48-s + 14·49-s − 4·52-s − 4·57-s + 14·61-s + 6·63-s − 64-s − 8·67-s + 4·73-s + 2·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 2.26·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.917·19-s − 1.30·21-s − 2/5·25-s − 0.192·27-s − 1.13·28-s − 0.718·31-s − 1/6·36-s + 0.328·37-s − 0.640·39-s + 1.37·43-s − 0.144·48-s + 2·49-s − 0.554·52-s − 0.529·57-s + 1.79·61-s + 0.755·63-s − 1/8·64-s − 0.977·67-s + 0.468·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450468 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450468 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.068931227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068931227\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 8 T + p T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
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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
−8.421614509791189962331797188341, −8.170186998359684926184999684069, −7.78998041890607523313507624663, −7.23765243145299859028804458710, −6.89889712699493438979408788803, −5.97283179263291363856126743387, −5.69701717715504038134108423394, −5.34000748952416938332431301890, −4.71381624069530879678754181621, −4.39385223217486358221500322685, −3.87836613121471117170905635911, −3.18774593412511201265229334705, −2.16731770562671379526911136441, −1.52333670144635464954712421481, −0.917223546182774170774519678844,
0.917223546182774170774519678844, 1.52333670144635464954712421481, 2.16731770562671379526911136441, 3.18774593412511201265229334705, 3.87836613121471117170905635911, 4.39385223217486358221500322685, 4.71381624069530879678754181621, 5.34000748952416938332431301890, 5.69701717715504038134108423394, 5.97283179263291363856126743387, 6.89889712699493438979408788803, 7.23765243145299859028804458710, 7.78998041890607523313507624663, 8.170186998359684926184999684069, 8.421614509791189962331797188341
