| L(s) = 1 | + 2-s − 1.50·3-s + 4-s + 5-s − 1.50·6-s − 3.09·7-s + 8-s − 0.725·9-s + 10-s − 0.743·11-s − 1.50·12-s + 13-s − 3.09·14-s − 1.50·15-s + 16-s − 2.93·17-s − 0.725·18-s + 2.92·19-s + 20-s + 4.66·21-s − 0.743·22-s + 1.11·23-s − 1.50·24-s + 25-s + 26-s + 5.61·27-s − 3.09·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.870·3-s + 0.5·4-s + 0.447·5-s − 0.615·6-s − 1.17·7-s + 0.353·8-s − 0.241·9-s + 0.316·10-s − 0.224·11-s − 0.435·12-s + 0.277·13-s − 0.827·14-s − 0.389·15-s + 0.250·16-s − 0.712·17-s − 0.170·18-s + 0.670·19-s + 0.223·20-s + 1.01·21-s − 0.158·22-s + 0.232·23-s − 0.307·24-s + 0.200·25-s + 0.196·26-s + 1.08·27-s − 0.585·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.696107505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.696107505\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 1.50T + 3T^{2} \) |
| 7 | \( 1 + 3.09T + 7T^{2} \) |
| 11 | \( 1 + 0.743T + 11T^{2} \) |
| 17 | \( 1 + 2.93T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 - 1.11T + 23T^{2} \) |
| 29 | \( 1 + 3.98T + 29T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 + 0.863T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 - 4.92T + 47T^{2} \) |
| 53 | \( 1 + 1.91T + 53T^{2} \) |
| 59 | \( 1 - 4.13T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 1.10T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 - 0.791T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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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
−8.430223276029948462310909086256, −7.43013013713703998879263539765, −6.53522706347277126340336158109, −6.28696121903791279193264706934, −5.45500535406707120632421650637, −4.94387674702724679678604409923, −3.82078906170958077869731881686, −3.07164114510982609991810437617, −2.16347997979863290079797463415, −0.67572658951951614739437465349,
0.67572658951951614739437465349, 2.16347997979863290079797463415, 3.07164114510982609991810437617, 3.82078906170958077869731881686, 4.94387674702724679678604409923, 5.45500535406707120632421650637, 6.28696121903791279193264706934, 6.53522706347277126340336158109, 7.43013013713703998879263539765, 8.430223276029948462310909086256
