L(s) = 1 | − 2-s − 3-s + 4-s + 3.17·5-s + 6-s − 4.61·7-s − 8-s + 9-s − 3.17·10-s − 0.697·11-s − 12-s + 2.07·13-s + 4.61·14-s − 3.17·15-s + 16-s + 17-s − 18-s − 4.21·19-s + 3.17·20-s + 4.61·21-s + 0.697·22-s + 1.76·23-s + 24-s + 5.06·25-s − 2.07·26-s − 27-s − 4.61·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.41·5-s + 0.408·6-s − 1.74·7-s − 0.353·8-s + 0.333·9-s − 1.00·10-s − 0.210·11-s − 0.288·12-s + 0.575·13-s + 1.23·14-s − 0.819·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 0.967·19-s + 0.709·20-s + 1.00·21-s + 0.148·22-s + 0.368·23-s + 0.204·24-s + 1.01·25-s − 0.406·26-s − 0.192·27-s − 0.871·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 - 3.17T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 0.697T + 11T^{2} \) |
| 13 | \( 1 - 2.07T + 13T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 23 | \( 1 - 1.76T + 23T^{2} \) |
| 29 | \( 1 + 2.67T + 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 37 | \( 1 - 6.98T + 37T^{2} \) |
| 41 | \( 1 + 9.67T + 41T^{2} \) |
| 43 | \( 1 + 0.221T + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 + 4.12T + 53T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 + 9.96T + 67T^{2} \) |
| 71 | \( 1 - 3.78T + 71T^{2} \) |
| 73 | \( 1 + 2.02T + 73T^{2} \) |
| 79 | \( 1 - 3.39T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 + 3.75T + 89T^{2} \) |
| 97 | \( 1 - 7.20T + 97T^{2} \) |
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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
−7.65791856517727707120503169966, −6.71252601677654233980778382305, −6.36212490836797642704348009551, −5.90902314951252538301697003726, −5.14467303882778625572383068487, −3.92155346225149546931876185181, −2.99184513728796674846617657926, −2.22944094666046863515652284800, −1.18134903739465789389831651384, 0,
1.18134903739465789389831651384, 2.22944094666046863515652284800, 2.99184513728796674846617657926, 3.92155346225149546931876185181, 5.14467303882778625572383068487, 5.90902314951252538301697003726, 6.36212490836797642704348009551, 6.71252601677654233980778382305, 7.65791856517727707120503169966