| L(s) = 1 | − 2-s + 3-s + 4-s + 0.480·5-s − 6-s + 0.765·7-s − 8-s + 9-s − 0.480·10-s − 1.12·11-s + 12-s − 13-s − 0.765·14-s + 0.480·15-s + 16-s + 0.131·17-s − 18-s + 7.23·19-s + 0.480·20-s + 0.765·21-s + 1.12·22-s − 6.47·23-s − 24-s − 4.76·25-s + 26-s + 27-s + 0.765·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.214·5-s − 0.408·6-s + 0.289·7-s − 0.353·8-s + 0.333·9-s − 0.152·10-s − 0.338·11-s + 0.288·12-s − 0.277·13-s − 0.204·14-s + 0.124·15-s + 0.250·16-s + 0.0318·17-s − 0.235·18-s + 1.65·19-s + 0.107·20-s + 0.167·21-s + 0.239·22-s − 1.35·23-s − 0.204·24-s − 0.953·25-s + 0.196·26-s + 0.192·27-s + 0.144·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 0.480T + 5T^{2} \) |
| 7 | \( 1 - 0.765T + 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 17 | \( 1 - 0.131T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 - 0.823T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 - 9.71T + 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 + 1.21T + 47T^{2} \) |
| 53 | \( 1 + 6.16T + 53T^{2} \) |
| 59 | \( 1 + 7.84T + 59T^{2} \) |
| 61 | \( 1 - 8.36T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 16.6T + 71T^{2} \) |
| 73 | \( 1 + 2.44T + 73T^{2} \) |
| 79 | \( 1 - 7.55T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−7.74969373064232638267315070516, −7.11713581824013246217840309899, −6.09353123158311738832950874937, −5.55789608892876491965656452513, −4.65645705847844051006551848208, −3.70649781194196757051952799354, −2.97815033207879842142089337210, −2.07270200427862947148767780617, −1.40145604728960189542194688654, 0,
1.40145604728960189542194688654, 2.07270200427862947148767780617, 2.97815033207879842142089337210, 3.70649781194196757051952799354, 4.65645705847844051006551848208, 5.55789608892876491965656452513, 6.09353123158311738832950874937, 7.11713581824013246217840309899, 7.74969373064232638267315070516
