| L(s) = 1 | − 2-s − 2.95·3-s + 4-s + 1.44·5-s + 2.95·6-s + 4.32·7-s − 8-s + 5.73·9-s − 1.44·10-s + 6.42·11-s − 2.95·12-s − 2.06·13-s − 4.32·14-s − 4.26·15-s + 16-s + 4.41·17-s − 5.73·18-s − 5.68·19-s + 1.44·20-s − 12.7·21-s − 6.42·22-s − 23-s + 2.95·24-s − 2.91·25-s + 2.06·26-s − 8.09·27-s + 4.32·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.70·3-s + 0.5·4-s + 0.645·5-s + 1.20·6-s + 1.63·7-s − 0.353·8-s + 1.91·9-s − 0.456·10-s + 1.93·11-s − 0.853·12-s − 0.572·13-s − 1.15·14-s − 1.10·15-s + 0.250·16-s + 1.07·17-s − 1.35·18-s − 1.30·19-s + 0.322·20-s − 2.79·21-s − 1.36·22-s − 0.208·23-s + 0.603·24-s − 0.583·25-s + 0.404·26-s − 1.55·27-s + 0.817·28-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
| \(L(1)\) |
\(\approx\) |
\(1.329143211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.329143211\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.95T + 3T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 - 4.32T + 7T^{2} \) |
| 11 | \( 1 - 6.42T + 11T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 + 5.68T + 19T^{2} \) |
| 29 | \( 1 - 1.15T + 29T^{2} \) |
| 31 | \( 1 + 4.03T + 31T^{2} \) |
| 37 | \( 1 - 8.86T + 37T^{2} \) |
| 41 | \( 1 - 8.32T + 41T^{2} \) |
| 43 | \( 1 - 8.00T + 43T^{2} \) |
| 47 | \( 1 - 7.79T + 47T^{2} \) |
| 53 | \( 1 + 7.90T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 1.27T + 61T^{2} \) |
| 67 | \( 1 + 5.02T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 5.36T + 73T^{2} \) |
| 79 | \( 1 - 5.85T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.907614051426241766174434222412, −7.36642914956988143462190306943, −6.53596675674477491776112009299, −5.92642560947227297457124685096, −5.51963973186130745168848682568, −4.46431548719623125423839007127, −4.09487885407652122121487724625, −2.23498094763895973308876160396, −1.46481072070268565826980027453, −0.837158698285452452181723123031,
0.837158698285452452181723123031, 1.46481072070268565826980027453, 2.23498094763895973308876160396, 4.09487885407652122121487724625, 4.46431548719623125423839007127, 5.51963973186130745168848682568, 5.92642560947227297457124685096, 6.53596675674477491776112009299, 7.36642914956988143462190306943, 7.907614051426241766174434222412
