L(s) = 1 | + 2-s + 4-s + 1.24·7-s + 8-s + 9-s − 0.445·11-s + 1.24·14-s + 16-s − 1.80·17-s + 18-s − 1.80·19-s − 0.445·22-s − 0.445·23-s + 25-s + 1.24·28-s − 1.80·29-s − 1.80·31-s + 32-s − 1.80·34-s + 36-s + 1.24·37-s − 1.80·38-s − 0.445·41-s + 1.24·43-s − 0.445·44-s − 0.445·46-s + 0.554·49-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 1.24·7-s + 8-s + 9-s − 0.445·11-s + 1.24·14-s + 16-s − 1.80·17-s + 18-s − 1.80·19-s − 0.445·22-s − 0.445·23-s + 25-s + 1.24·28-s − 1.80·29-s − 1.80·31-s + 32-s − 1.80·34-s + 36-s + 1.24·37-s − 1.80·38-s − 0.445·41-s + 1.24·43-s − 0.445·44-s − 0.445·46-s + 0.554·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.438674841\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.438674841\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 1.24T + T^{2} \) |
| 11 | \( 1 + 0.445T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.80T + T^{2} \) |
| 19 | \( 1 + 1.80T + T^{2} \) |
| 23 | \( 1 + 0.445T + T^{2} \) |
| 29 | \( 1 + 1.80T + T^{2} \) |
| 31 | \( 1 + 1.80T + T^{2} \) |
| 37 | \( 1 - 1.24T + T^{2} \) |
| 41 | \( 1 + 0.445T + T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.445T + T^{2} \) |
| 59 | \( 1 - 1.24T + T^{2} \) |
| 61 | \( 1 + 0.445T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.24T + T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.287070212578754214536198914897, −8.417624210548191230706640473169, −7.57219883163814118285651831258, −6.96286850243061467356775829999, −6.09730481407786616323716315548, −5.13081514742221194470586310891, −4.40157628475435466194178330581, −3.93671003577164193694341772594, −2.30140953731774952545730097360, −1.79916743201281788443203136104,
1.79916743201281788443203136104, 2.30140953731774952545730097360, 3.93671003577164193694341772594, 4.40157628475435466194178330581, 5.13081514742221194470586310891, 6.09730481407786616323716315548, 6.96286850243061467356775829999, 7.57219883163814118285651831258, 8.417624210548191230706640473169, 9.287070212578754214536198914897