L(s) = 1 | − 2-s + 2.79·3-s + 4-s − 2.21·5-s − 2.79·6-s + 4.37·7-s − 8-s + 4.82·9-s + 2.21·10-s + 0.826·11-s + 2.79·12-s + 4.83·13-s − 4.37·14-s − 6.20·15-s + 16-s − 3.78·17-s − 4.82·18-s − 19-s − 2.21·20-s + 12.2·21-s − 0.826·22-s − 4.39·23-s − 2.79·24-s − 0.0846·25-s − 4.83·26-s + 5.09·27-s + 4.37·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.61·3-s + 0.5·4-s − 0.991·5-s − 1.14·6-s + 1.65·7-s − 0.353·8-s + 1.60·9-s + 0.701·10-s + 0.249·11-s + 0.807·12-s + 1.34·13-s − 1.17·14-s − 1.60·15-s + 0.250·16-s − 0.919·17-s − 1.13·18-s − 0.229·19-s − 0.495·20-s + 2.67·21-s − 0.176·22-s − 0.916·23-s − 0.570·24-s − 0.0169·25-s − 0.948·26-s + 0.981·27-s + 0.827·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.088471979\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.088471979\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 - 0.826T + 11T^{2} \) |
| 13 | \( 1 - 4.83T + 13T^{2} \) |
| 17 | \( 1 + 3.78T + 17T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 + 0.511T + 31T^{2} \) |
| 37 | \( 1 + 0.132T + 37T^{2} \) |
| 41 | \( 1 - 9.41T + 41T^{2} \) |
| 43 | \( 1 + 5.37T + 43T^{2} \) |
| 47 | \( 1 - 8.73T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 8.88T + 61T^{2} \) |
| 67 | \( 1 - 4.15T + 67T^{2} \) |
| 71 | \( 1 + 5.11T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 7.06T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 4.55T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−8.115343226300126247298686923584, −7.55600761298696761961709618656, −6.83979413672850924059698112124, −5.90789972615129831451122081701, −4.68176893872899184877649445834, −4.05736312754474012776527372512, −3.56787139302592010303620095606, −2.43889157587680248504333827881, −1.86160757841868904552193606472, −0.940859947194092860385011508023,
0.940859947194092860385011508023, 1.86160757841868904552193606472, 2.43889157587680248504333827881, 3.56787139302592010303620095606, 4.05736312754474012776527372512, 4.68176893872899184877649445834, 5.90789972615129831451122081701, 6.83979413672850924059698112124, 7.55600761298696761961709618656, 8.115343226300126247298686923584