| L(s) = 1 | − 2.57·2-s − 3-s + 4.64·4-s + 2.57·6-s − 4.69·7-s − 6.81·8-s + 9-s − 3.11·11-s − 4.64·12-s + 5.07·13-s + 12.1·14-s + 8.28·16-s − 1.40·17-s − 2.57·18-s − 3.76·19-s + 4.69·21-s + 8.03·22-s − 5.71·23-s + 6.81·24-s − 13.0·26-s − 27-s − 21.8·28-s + 29-s − 2.23·31-s − 7.73·32-s + 3.11·33-s + 3.60·34-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.577·3-s + 2.32·4-s + 1.05·6-s − 1.77·7-s − 2.41·8-s + 0.333·9-s − 0.939·11-s − 1.34·12-s + 1.40·13-s + 3.23·14-s + 2.07·16-s − 0.339·17-s − 0.607·18-s − 0.863·19-s + 1.02·21-s + 1.71·22-s − 1.19·23-s + 1.39·24-s − 2.56·26-s − 0.192·27-s − 4.12·28-s + 0.185·29-s − 0.400·31-s − 1.36·32-s + 0.542·33-s + 0.619·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1907559889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1907559889\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 7 | \( 1 + 4.69T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 - 5.07T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 + 5.71T + 23T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 + 5.79T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 8.89T + 43T^{2} \) |
| 47 | \( 1 + 3.62T + 47T^{2} \) |
| 53 | \( 1 - 0.948T + 53T^{2} \) |
| 59 | \( 1 + 8.53T + 59T^{2} \) |
| 61 | \( 1 + 6.21T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 + 7.08T + 73T^{2} \) |
| 79 | \( 1 - 7.31T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 7.98T + 89T^{2} \) |
| 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
−9.084181927340036589792838591766, −8.464313309952598701659844387715, −7.67458491261962899226467640778, −6.65948518864007495867065960067, −6.39905280928136446165861319993, −5.58634779304436625833057777724, −3.90458636026905220803120738912, −2.92319217425504110822603056912, −1.80901281345292221085470937463, −0.36888289924507259664087467196,
0.36888289924507259664087467196, 1.80901281345292221085470937463, 2.92319217425504110822603056912, 3.90458636026905220803120738912, 5.58634779304436625833057777724, 6.39905280928136446165861319993, 6.65948518864007495867065960067, 7.67458491261962899226467640778, 8.464313309952598701659844387715, 9.084181927340036589792838591766
