| L(s) = 1 | + 2·4-s + 3·16-s − 49-s + 4·64-s − 4·79-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s − 49-s + 4·64-s − 4·79-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.976546573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.976546573\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$ | \( ( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.777662703165187287566671656296, −9.751939668581427896467730196113, −8.785094445161926091883884528579, −8.696821163695744809947302107927, −8.120982520058901001726358376490, −7.69533806277893102285417128286, −7.25543411897011803892344680586, −7.20254116341639988099969785528, −6.42567958369940700772505248296, −6.41633951449973542666162314434, −5.72494990932537743023290213341, −5.62711434928389653058091435628, −4.85669050903742721122450969610, −4.44484030762349950054090385695, −3.53510049815431303241984854106, −3.47715774174445045269970651000, −2.61732237927340724392963698213, −2.49888726773389951467309249421, −1.66337124634595529707566199574, −1.27033692199437885483573765158,
1.27033692199437885483573765158, 1.66337124634595529707566199574, 2.49888726773389951467309249421, 2.61732237927340724392963698213, 3.47715774174445045269970651000, 3.53510049815431303241984854106, 4.44484030762349950054090385695, 4.85669050903742721122450969610, 5.62711434928389653058091435628, 5.72494990932537743023290213341, 6.41633951449973542666162314434, 6.42567958369940700772505248296, 7.20254116341639988099969785528, 7.25543411897011803892344680586, 7.69533806277893102285417128286, 8.120982520058901001726358376490, 8.696821163695744809947302107927, 8.785094445161926091883884528579, 9.751939668581427896467730196113, 9.777662703165187287566671656296
