| L(s) = 1 | + 2-s + 1.85·3-s + 4-s + 3.38·5-s + 1.85·6-s + 1.90·7-s + 8-s + 0.439·9-s + 3.38·10-s − 11-s + 1.85·12-s + 5.80·13-s + 1.90·14-s + 6.27·15-s + 16-s − 2.03·17-s + 0.439·18-s − 0.936·19-s + 3.38·20-s + 3.53·21-s − 22-s + 5.36·23-s + 1.85·24-s + 6.46·25-s + 5.80·26-s − 4.74·27-s + 1.90·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.07·3-s + 0.5·4-s + 1.51·5-s + 0.757·6-s + 0.719·7-s + 0.353·8-s + 0.146·9-s + 1.07·10-s − 0.301·11-s + 0.535·12-s + 1.61·13-s + 0.508·14-s + 1.62·15-s + 0.250·16-s − 0.492·17-s + 0.103·18-s − 0.214·19-s + 0.757·20-s + 0.770·21-s − 0.213·22-s + 1.11·23-s + 0.378·24-s + 1.29·25-s + 1.13·26-s − 0.913·27-s + 0.359·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.577842201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.577842201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 1.85T + 3T^{2} \) |
| 5 | \( 1 - 3.38T + 5T^{2} \) |
| 7 | \( 1 - 1.90T + 7T^{2} \) |
| 13 | \( 1 - 5.80T + 13T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 + 0.936T + 19T^{2} \) |
| 23 | \( 1 - 5.36T + 23T^{2} \) |
| 29 | \( 1 + 1.64T + 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 - 4.49T + 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 7.86T + 59T^{2} \) |
| 61 | \( 1 + 5.64T + 61T^{2} \) |
| 67 | \( 1 + 7.90T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 + 8.89T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 1.98T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.407511283635891952738247311375, −7.79380003214261411301376492017, −6.67313397558848978144559457917, −6.17070791163194960095210652964, −5.35116840574935147604290207132, −4.74253042805140248212249052717, −3.57638768436113385755947904076, −2.98709372318200119531076427393, −1.96903808175655098500631931557, −1.53493140110409542318763490956,
1.53493140110409542318763490956, 1.96903808175655098500631931557, 2.98709372318200119531076427393, 3.57638768436113385755947904076, 4.74253042805140248212249052717, 5.35116840574935147604290207132, 6.17070791163194960095210652964, 6.67313397558848978144559457917, 7.79380003214261411301376492017, 8.407511283635891952738247311375
