L(s) = 1 | − 3·3-s + 14·5-s + 7·7-s + 9·9-s − 4·11-s + 54·13-s − 42·15-s − 14·17-s − 92·19-s − 21·21-s + 152·23-s + 71·25-s − 27·27-s − 106·29-s + 144·31-s + 12·33-s + 98·35-s + 158·37-s − 162·39-s − 390·41-s + 508·43-s + 126·45-s + 528·47-s + 49·49-s + 42·51-s + 606·53-s − 56·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.25·5-s + 0.377·7-s + 1/3·9-s − 0.109·11-s + 1.15·13-s − 0.722·15-s − 0.199·17-s − 1.11·19-s − 0.218·21-s + 1.37·23-s + 0.567·25-s − 0.192·27-s − 0.678·29-s + 0.834·31-s + 0.0633·33-s + 0.473·35-s + 0.702·37-s − 0.665·39-s − 1.48·41-s + 1.80·43-s + 0.417·45-s + 1.63·47-s + 1/7·49-s + 0.115·51-s + 1.57·53-s − 0.137·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.113749705\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.113749705\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
good | 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 - 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 158 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 - 508 T + p^{3} T^{2} \) |
| 47 | \( 1 - 528 T + p^{3} T^{2} \) |
| 53 | \( 1 - 606 T + p^{3} T^{2} \) |
| 59 | \( 1 - 364 T + p^{3} T^{2} \) |
| 61 | \( 1 - 678 T + p^{3} T^{2} \) |
| 67 | \( 1 + 844 T + p^{3} T^{2} \) |
| 71 | \( 1 - 8 T + p^{3} T^{2} \) |
| 73 | \( 1 + 422 T + p^{3} T^{2} \) |
| 79 | \( 1 + 384 T + p^{3} T^{2} \) |
| 83 | \( 1 - 548 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1502 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.91034244295928475835276143207, −10.43212043408376003827637800441, −9.274282088136640568981252223799, −8.491250810901165225797197107152, −7.05145532702476381667586986961, −6.10300272441339421304747494026, −5.39808945544984313617954476785, −4.14545007433564799527712388567, −2.38473203943799878251476206186, −1.09898506836191065390739121338,
1.09898506836191065390739121338, 2.38473203943799878251476206186, 4.14545007433564799527712388567, 5.39808945544984313617954476785, 6.10300272441339421304747494026, 7.05145532702476381667586986961, 8.491250810901165225797197107152, 9.274282088136640568981252223799, 10.43212043408376003827637800441, 10.91034244295928475835276143207