| L(s) = 1 | − 1.69·2-s + 0.407·3-s + 0.869·4-s + 5-s − 0.690·6-s + 2.45·7-s + 1.91·8-s − 2.83·9-s − 1.69·10-s − 11-s + 0.354·12-s + 3.09·13-s − 4.15·14-s + 0.407·15-s − 4.98·16-s + 4.42·17-s + 4.79·18-s − 1.16·19-s + 0.869·20-s + 1.00·21-s + 1.69·22-s − 0.0825·23-s + 0.781·24-s + 25-s − 5.24·26-s − 2.37·27-s + 2.13·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.235·3-s + 0.434·4-s + 0.447·5-s − 0.282·6-s + 0.927·7-s + 0.677·8-s − 0.944·9-s − 0.535·10-s − 0.301·11-s + 0.102·12-s + 0.858·13-s − 1.11·14-s + 0.105·15-s − 1.24·16-s + 1.07·17-s + 1.13·18-s − 0.268·19-s + 0.194·20-s + 0.218·21-s + 0.361·22-s − 0.0172·23-s + 0.159·24-s + 0.200·25-s − 1.02·26-s − 0.457·27-s + 0.402·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.232485264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.232485264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 3 | \( 1 - 0.407T + 3T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 + 0.0825T + 23T^{2} \) |
| 29 | \( 1 - 4.64T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + 0.193T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 + 8.04T + 59T^{2} \) |
| 61 | \( 1 - 7.78T + 61T^{2} \) |
| 67 | \( 1 - 9.98T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 79 | \( 1 + 7.03T + 79T^{2} \) |
| 83 | \( 1 - 9.56T + 83T^{2} \) |
| 89 | \( 1 - 9.83T + 89T^{2} \) |
| 97 | \( 1 - 3.22T + 97T^{2} \) |
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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
−8.541224075588013928926364163780, −7.85814448113057429542010207828, −7.46664361109898124444794970248, −6.24581045424175419275715184227, −5.58480551909094160997016028304, −4.77425929792583888823725691200, −3.75699413029378794611943807206, −2.63166105647109223030986434425, −1.72404294385190850205402914330, −0.791061600939441994317132523311,
0.791061600939441994317132523311, 1.72404294385190850205402914330, 2.63166105647109223030986434425, 3.75699413029378794611943807206, 4.77425929792583888823725691200, 5.58480551909094160997016028304, 6.24581045424175419275715184227, 7.46664361109898124444794970248, 7.85814448113057429542010207828, 8.541224075588013928926364163780
