| L(s) = 1 | + 4-s + 0.347·7-s + 9-s + 1.53·11-s + 16-s − 1.87·17-s + 25-s + 0.347·28-s − 29-s − 31-s + 36-s − 43-s + 1.53·44-s − 0.879·49-s − 1.87·61-s + 0.347·63-s + 64-s − 1.87·68-s − 1.87·71-s + 0.532·77-s + 1.53·79-s + 81-s + 1.53·97-s + 1.53·99-s + 100-s + 1.53·103-s − 1.87·107-s + ⋯ |
| L(s) = 1 | + 4-s + 0.347·7-s + 9-s + 1.53·11-s + 16-s − 1.87·17-s + 25-s + 0.347·28-s − 29-s − 31-s + 36-s − 43-s + 1.53·44-s − 0.879·49-s − 1.87·61-s + 0.347·63-s + 64-s − 1.87·68-s − 1.87·71-s + 0.532·77-s + 1.53·79-s + 81-s + 1.53·97-s + 1.53·99-s + 100-s + 1.53·103-s − 1.87·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3307 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3307 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.890861083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.890861083\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3307 | \( 1+O(T) \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.347T + T^{2} \) |
| 11 | \( 1 - 1.53T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.87T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.87T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.53T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.53T + T^{2} \) |
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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
−8.993056089263521723865065148420, −7.940612000799010608330727948979, −7.09289247972762028394469081746, −6.72254230929174578562701846515, −6.07235780267538217812842975771, −4.87673215564131980366530419122, −4.15035818942149462310252870186, −3.29451797226905371365679376390, −2.01226473916890061342253667050, −1.46169696116498875555947820323,
1.46169696116498875555947820323, 2.01226473916890061342253667050, 3.29451797226905371365679376390, 4.15035818942149462310252870186, 4.87673215564131980366530419122, 6.07235780267538217812842975771, 6.72254230929174578562701846515, 7.09289247972762028394469081746, 7.940612000799010608330727948979, 8.993056089263521723865065148420
