| L(s) = 1 | + 1.62·2-s + 3-s + 0.644·4-s + 1.67·5-s + 1.62·6-s − 2.20·8-s + 9-s + 2.72·10-s + 2.72·11-s + 0.644·12-s − 2.59·13-s + 1.67·15-s − 4.87·16-s + 2.49·17-s + 1.62·18-s − 0.765·19-s + 1.07·20-s + 4.43·22-s + 1.55·23-s − 2.20·24-s − 2.19·25-s − 4.21·26-s + 27-s + 9.11·29-s + 2.72·30-s + 8.26·31-s − 3.51·32-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.322·4-s + 0.748·5-s + 0.663·6-s − 0.779·8-s + 0.333·9-s + 0.860·10-s + 0.821·11-s + 0.186·12-s − 0.718·13-s + 0.432·15-s − 1.21·16-s + 0.605·17-s + 0.383·18-s − 0.175·19-s + 0.241·20-s + 0.944·22-s + 0.323·23-s − 0.449·24-s − 0.439·25-s − 0.826·26-s + 0.192·27-s + 1.69·29-s + 0.496·30-s + 1.48·31-s − 0.621·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.097285518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.097285518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.62T + 2T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 + 2.59T + 13T^{2} \) |
| 17 | \( 1 - 2.49T + 17T^{2} \) |
| 19 | \( 1 + 0.765T + 19T^{2} \) |
| 23 | \( 1 - 1.55T + 23T^{2} \) |
| 29 | \( 1 - 9.11T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 + 3.21T + 37T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 5.81T + 47T^{2} \) |
| 53 | \( 1 + 4.59T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 + 0.867T + 61T^{2} \) |
| 67 | \( 1 - 3.89T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 2.23T + 73T^{2} \) |
| 79 | \( 1 - 8.10T + 79T^{2} \) |
| 83 | \( 1 - 8.58T + 83T^{2} \) |
| 89 | \( 1 - 1.25T + 89T^{2} \) |
| 97 | \( 1 - 8.55T + 97T^{2} \) |
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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
−8.056394254929366769252105732127, −7.21529169690205461757379178375, −6.34925252594322355516343124740, −5.99718980121794281601192869205, −5.00000437322537542679478361456, −4.50283409876726704520642859282, −3.70236307519436342687385386726, −2.86479688708274493416816186122, −2.26402994211093032574895750899, −1.00741684846325318275026327863,
1.00741684846325318275026327863, 2.26402994211093032574895750899, 2.86479688708274493416816186122, 3.70236307519436342687385386726, 4.50283409876726704520642859282, 5.00000437322537542679478361456, 5.99718980121794281601192869205, 6.34925252594322355516343124740, 7.21529169690205461757379178375, 8.056394254929366769252105732127
