| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 1.07·7-s − 8-s + 9-s + 6.34·11-s + 12-s + 3.41·13-s + 1.07·14-s + 16-s − 5.41·17-s − 18-s + 19-s − 1.07·21-s − 6.34·22-s + 6.34·23-s − 24-s − 3.41·26-s + 27-s − 1.07·28-s − 0.340·29-s + 1.07·31-s − 32-s + 6.34·33-s + 5.41·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.407·7-s − 0.353·8-s + 0.333·9-s + 1.91·11-s + 0.288·12-s + 0.948·13-s + 0.288·14-s + 0.250·16-s − 1.31·17-s − 0.235·18-s + 0.229·19-s − 0.235·21-s − 1.35·22-s + 1.32·23-s − 0.204·24-s − 0.670·26-s + 0.192·27-s − 0.203·28-s − 0.0631·29-s + 0.193·31-s − 0.176·32-s + 1.10·33-s + 0.929·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.863181829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.863181829\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 1.07T + 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 23 | \( 1 - 6.34T + 23T^{2} \) |
| 29 | \( 1 + 0.340T + 29T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 - 7.60T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 6.34T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 0.738T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 - 2.83T + 67T^{2} \) |
| 71 | \( 1 + 2.83T + 71T^{2} \) |
| 73 | \( 1 - 6.83T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 - 0.894T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.849088195923845040204333572579, −8.329166307582065078744952634109, −7.22359857185165900304324180377, −6.61062425107664451883093656253, −6.15441964035426648046034271803, −4.73080072424621857079550472221, −3.81615158112852592072856875039, −3.10806321221570367114970203116, −1.89636841695441668962170976633, −0.969173666826125565904573100662,
0.969173666826125565904573100662, 1.89636841695441668962170976633, 3.10806321221570367114970203116, 3.81615158112852592072856875039, 4.73080072424621857079550472221, 6.15441964035426648046034271803, 6.61062425107664451883093656253, 7.22359857185165900304324180377, 8.329166307582065078744952634109, 8.849088195923845040204333572579
