L(s) = 1 | + 2-s − 1.81·3-s + 4-s + 5-s − 1.81·6-s − 3.01·7-s + 8-s + 0.297·9-s + 10-s − 5.40·11-s − 1.81·12-s − 2.18·13-s − 3.01·14-s − 1.81·15-s + 16-s + 6.31·17-s + 0.297·18-s − 6.90·19-s + 20-s + 5.46·21-s − 5.40·22-s − 7.90·23-s − 1.81·24-s + 25-s − 2.18·26-s + 4.90·27-s − 3.01·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.04·3-s + 0.5·4-s + 0.447·5-s − 0.741·6-s − 1.13·7-s + 0.353·8-s + 0.0991·9-s + 0.316·10-s − 1.62·11-s − 0.524·12-s − 0.605·13-s − 0.804·14-s − 0.468·15-s + 0.250·16-s + 1.53·17-s + 0.0701·18-s − 1.58·19-s + 0.223·20-s + 1.19·21-s − 1.15·22-s − 1.64·23-s − 0.370·24-s + 0.200·25-s − 0.428·26-s + 0.944·27-s − 0.569·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.143495848\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.143495848\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 1.81T + 3T^{2} \) |
| 7 | \( 1 + 3.01T + 7T^{2} \) |
| 11 | \( 1 + 5.40T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 - 6.31T + 17T^{2} \) |
| 19 | \( 1 + 6.90T + 19T^{2} \) |
| 23 | \( 1 + 7.90T + 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 9.90T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 6.09T + 53T^{2} \) |
| 59 | \( 1 - 0.568T + 59T^{2} \) |
| 61 | \( 1 - 6.77T + 61T^{2} \) |
| 67 | \( 1 + 3.56T + 67T^{2} \) |
| 71 | \( 1 - 4.92T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 3.75T + 89T^{2} \) |
| 97 | \( 1 + 9.37T + 97T^{2} \) |
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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
−8.034353558219242193170682899736, −7.78148268311898878075958309160, −6.36866336548491924680478984632, −6.28954118706166087586941821233, −5.52619388850144274084058738884, −4.91223928746852931344261045737, −3.98051899299622682524171365039, −2.86097045651172218795887882177, −2.30666301981735120787773257469, −0.54213002192564223195157686862,
0.54213002192564223195157686862, 2.30666301981735120787773257469, 2.86097045651172218795887882177, 3.98051899299622682524171365039, 4.91223928746852931344261045737, 5.52619388850144274084058738884, 6.28954118706166087586941821233, 6.36866336548491924680478984632, 7.78148268311898878075958309160, 8.034353558219242193170682899736