L(s) = 1 | + 2-s + 2.40·3-s + 4-s − 2.20·5-s + 2.40·6-s + 8-s + 2.79·9-s − 2.20·10-s − 1.18·11-s + 2.40·12-s + 1.24·13-s − 5.31·15-s + 16-s + 5.96·17-s + 2.79·18-s + 2.19·19-s − 2.20·20-s − 1.18·22-s + 7.74·23-s + 2.40·24-s − 0.125·25-s + 1.24·26-s − 0.497·27-s − 29-s − 5.31·30-s + 9.31·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.38·3-s + 0.5·4-s − 0.987·5-s + 0.982·6-s + 0.353·8-s + 0.931·9-s − 0.698·10-s − 0.358·11-s + 0.694·12-s + 0.344·13-s − 1.37·15-s + 0.250·16-s + 1.44·17-s + 0.658·18-s + 0.503·19-s − 0.493·20-s − 0.253·22-s + 1.61·23-s + 0.491·24-s − 0.0250·25-s + 0.243·26-s − 0.0956·27-s − 0.185·29-s − 0.970·30-s + 1.67·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.263037497\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.263037497\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 11 | \( 1 + 1.18T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 - 5.96T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 - 7.74T + 23T^{2} \) |
| 31 | \( 1 - 9.31T + 31T^{2} \) |
| 37 | \( 1 + 6.90T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 + 7.97T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 4.61T + 71T^{2} \) |
| 73 | \( 1 - 0.0394T + 73T^{2} \) |
| 79 | \( 1 + 2.42T + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 + 7.26T + 89T^{2} \) |
| 97 | \( 1 - 3.23T + 97T^{2} \) |
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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
−8.531264035881866987204464976793, −7.977322256020306115142058995191, −7.49230941532195937429480600804, −6.68083128926402428858037913056, −5.51873412781968226757053157172, −4.72938638812503851432820357597, −3.70373708554858429577736277636, −3.29100945717872858758067622425, −2.54273354544628899250003261932, −1.16073713084145760465515114173,
1.16073713084145760465515114173, 2.54273354544628899250003261932, 3.29100945717872858758067622425, 3.70373708554858429577736277636, 4.72938638812503851432820357597, 5.51873412781968226757053157172, 6.68083128926402428858037913056, 7.49230941532195937429480600804, 7.977322256020306115142058995191, 8.531264035881866987204464976793