| L(s) = 1 | + 2-s + 4-s − 0.445·7-s + 8-s + 9-s − 1.80·11-s − 0.445·14-s + 16-s + 1.24·17-s + 18-s + 1.24·19-s − 1.80·22-s − 1.80·23-s + 25-s − 0.445·28-s + 1.24·29-s + 1.24·31-s + 32-s + 1.24·34-s + 36-s − 0.445·37-s + 1.24·38-s − 1.80·41-s − 0.445·43-s − 1.80·44-s − 1.80·46-s − 0.801·49-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 0.445·7-s + 8-s + 9-s − 1.80·11-s − 0.445·14-s + 16-s + 1.24·17-s + 18-s + 1.24·19-s − 1.80·22-s − 1.80·23-s + 25-s − 0.445·28-s + 1.24·29-s + 1.24·31-s + 32-s + 1.24·34-s + 36-s − 0.445·37-s + 1.24·38-s − 1.80·41-s − 0.445·43-s − 1.80·44-s − 1.80·46-s − 0.801·49-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.189949895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189949895\) |
| \(L(1)\) |
|
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 | \( 1 + 0.445T + T^{2} \) |
| 11 | \( 1 + 1.80T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.24T + T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 - 1.24T + T^{2} \) |
| 31 | \( 1 - 1.24T + T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 + 0.445T + T^{2} \) |
| 61 | \( 1 + 1.80T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.445T + T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 0.445T + T^{2} \) |
| 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
−9.787816506003684698192300407786, −8.072614514188752656720750219597, −7.82135070647735025724817902482, −6.85446611413567599809651854315, −6.10152828771663925452510106786, −5.13453697736513187885014052533, −4.68071065382005982257663483019, −3.38788895230886524496922558945, −2.85265783201785653389638142199, −1.51044338409629372437500213854,
1.51044338409629372437500213854, 2.85265783201785653389638142199, 3.38788895230886524496922558945, 4.68071065382005982257663483019, 5.13453697736513187885014052533, 6.10152828771663925452510106786, 6.85446611413567599809651854315, 7.82135070647735025724817902482, 8.072614514188752656720750219597, 9.787816506003684698192300407786
