| L(s) = 1 | − 2·13-s − 6·17-s − 14·37-s − 16·41-s + 18·53-s − 24·61-s + 22·73-s − 9·81-s − 26·97-s − 4·101-s + 2·113-s + 22·121-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 + ⋯ |
| L(s) = 1 | − 0.554·13-s − 1.45·17-s − 2.30·37-s − 2.49·41-s + 2.47·53-s − 3.07·61-s + 2.57·73-s − 81-s − 2.63·97-s − 0.398·101-s + 0.188·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9195427721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9195427721\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.767903981935925295708492327207, −9.158047673976675898491476918859, −8.687258232474921851433377090122, −8.505857169049932365334168389233, −8.226234801848473967582035547826, −7.39847823269285495455918527513, −7.25980775975196894870560459596, −6.77304054145758788074451628387, −6.55043374149371581466503007737, −5.95840057996330046255882319759, −5.44955610623167730756439373762, −4.95607936423897530364484505212, −4.81645057521060485154422430691, −4.04574733067508784466632777220, −3.76485928125883358608548295484, −3.10195229467936784608381047703, −2.62941979123255925439395993849, −1.92297272797417177597877392857, −1.58776063976500662580433775881, −0.35694884930754046654729483871,
0.35694884930754046654729483871, 1.58776063976500662580433775881, 1.92297272797417177597877392857, 2.62941979123255925439395993849, 3.10195229467936784608381047703, 3.76485928125883358608548295484, 4.04574733067508784466632777220, 4.81645057521060485154422430691, 4.95607936423897530364484505212, 5.44955610623167730756439373762, 5.95840057996330046255882319759, 6.55043374149371581466503007737, 6.77304054145758788074451628387, 7.25980775975196894870560459596, 7.39847823269285495455918527513, 8.226234801848473967582035547826, 8.505857169049932365334168389233, 8.687258232474921851433377090122, 9.158047673976675898491476918859, 9.767903981935925295708492327207
