L(s) = 1 | − 2.40·2-s + 3-s + 3.78·4-s + 2.94·5-s − 2.40·6-s + 2.89·7-s − 4.28·8-s + 9-s − 7.08·10-s − 2.53·11-s + 3.78·12-s + 5.47·13-s − 6.95·14-s + 2.94·15-s + 2.72·16-s − 3.33·17-s − 2.40·18-s − 5.49·19-s + 11.1·20-s + 2.89·21-s + 6.09·22-s + 23-s − 4.28·24-s + 3.68·25-s − 13.1·26-s + 27-s + 10.9·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 0.577·3-s + 1.89·4-s + 1.31·5-s − 0.981·6-s + 1.09·7-s − 1.51·8-s + 0.333·9-s − 2.23·10-s − 0.764·11-s + 1.09·12-s + 1.51·13-s − 1.85·14-s + 0.760·15-s + 0.682·16-s − 0.809·17-s − 0.566·18-s − 1.25·19-s + 2.49·20-s + 0.631·21-s + 1.29·22-s + 0.208·23-s − 0.873·24-s + 0.736·25-s − 2.58·26-s + 0.192·27-s + 2.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.460598450\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.460598450\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 + 5.49T + 19T^{2} \) |
| 31 | \( 1 - 5.60T + 31T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 + 9.72T + 41T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 1.45T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 5.14T + 67T^{2} \) |
| 71 | \( 1 + 3.43T + 71T^{2} \) |
| 73 | \( 1 - 6.90T + 73T^{2} \) |
| 79 | \( 1 + 5.51T + 79T^{2} \) |
| 83 | \( 1 - 0.148T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
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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
−8.965462996617056793310101496764, −8.416542575843597353224671158338, −8.157690826589865858359629546045, −6.96146278152118686424130854437, −6.33772637510591035481084958327, −5.37086521550289941974207112571, −4.18218833500937853783483411244, −2.53287090058433394451671207744, −2.00924852733625435686250778000, −1.08512967675122435718368959560,
1.08512967675122435718368959560, 2.00924852733625435686250778000, 2.53287090058433394451671207744, 4.18218833500937853783483411244, 5.37086521550289941974207112571, 6.33772637510591035481084958327, 6.96146278152118686424130854437, 8.157690826589865858359629546045, 8.416542575843597353224671158338, 8.965462996617056793310101496764