| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 10-s − 12-s − 15-s + 16-s + 20-s − 23-s − 24-s + 25-s + 27-s − 29-s − 30-s + 32-s + 40-s − 41-s − 43-s − 46-s + 2·47-s − 48-s + 50-s + 54-s − 58-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 10-s − 12-s − 15-s + 16-s + 20-s − 23-s − 24-s + 25-s + 27-s − 29-s − 30-s + 32-s + 40-s − 41-s − 43-s − 46-s + 2·47-s − 48-s + 50-s + 54-s − 58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.571586136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.571586136\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−10.48837682248824403502684089034, −9.680813168635236503404947597466, −8.495183693591226191025698679234, −7.27531478283278847421681503952, −6.43243409833182156386346849987, −5.75648264086940638318027988780, −5.24625313249891987092602066361, −4.19287236100256119260879849162, −2.87906353369821104840930910642, −1.68484701000591052647668112679,
1.68484701000591052647668112679, 2.87906353369821104840930910642, 4.19287236100256119260879849162, 5.24625313249891987092602066361, 5.75648264086940638318027988780, 6.43243409833182156386346849987, 7.27531478283278847421681503952, 8.495183693591226191025698679234, 9.680813168635236503404947597466, 10.48837682248824403502684089034
