L(s) = 1 | + 2-s − 1.52·3-s + 4-s + 1.80·5-s − 1.52·6-s + 2.28·7-s + 8-s − 0.665·9-s + 1.80·10-s + 6.02·11-s − 1.52·12-s − 1.69·13-s + 2.28·14-s − 2.76·15-s + 16-s + 4.34·17-s − 0.665·18-s + 7.78·19-s + 1.80·20-s − 3.48·21-s + 6.02·22-s − 23-s − 1.52·24-s − 1.72·25-s − 1.69·26-s + 5.60·27-s + 2.28·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.882·3-s + 0.5·4-s + 0.809·5-s − 0.623·6-s + 0.862·7-s + 0.353·8-s − 0.221·9-s + 0.572·10-s + 1.81·11-s − 0.441·12-s − 0.470·13-s + 0.609·14-s − 0.714·15-s + 0.250·16-s + 1.05·17-s − 0.156·18-s + 1.78·19-s + 0.404·20-s − 0.760·21-s + 1.28·22-s − 0.208·23-s − 0.311·24-s − 0.344·25-s − 0.332·26-s + 1.07·27-s + 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.683207228\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.683207228\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 1.52T + 3T^{2} \) |
| 5 | \( 1 - 1.80T + 5T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 - 6.02T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 - 4.34T + 17T^{2} \) |
| 19 | \( 1 - 7.78T + 19T^{2} \) |
| 29 | \( 1 - 2.33T + 29T^{2} \) |
| 31 | \( 1 - 4.63T + 31T^{2} \) |
| 37 | \( 1 + 3.98T + 37T^{2} \) |
| 41 | \( 1 + 6.85T + 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 - 0.341T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 9.39T + 61T^{2} \) |
| 67 | \( 1 - 9.80T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 + 0.667T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 1.05T + 89T^{2} \) |
| 97 | \( 1 + 8.59T + 97T^{2} \) |
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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
−7.923540715587937494385848126466, −7.12185777491704402139732686979, −6.42184773392858986824826406100, −5.83999119597267055742191319307, −5.24231887032783260173552807589, −4.73006735350275286900056699355, −3.72401197162816680849785177894, −2.90197735405548377545279558369, −1.66548493548728353119942155721, −1.07267455446090966587748279701,
1.07267455446090966587748279701, 1.66548493548728353119942155721, 2.90197735405548377545279558369, 3.72401197162816680849785177894, 4.73006735350275286900056699355, 5.24231887032783260173552807589, 5.83999119597267055742191319307, 6.42184773392858986824826406100, 7.12185777491704402139732686979, 7.923540715587937494385848126466