| L(s) = 1 | − 2.64·2-s + 3-s + 5.00·4-s − 2.64·6-s + 3.64·7-s − 7.93·8-s + 9-s + 5.64·11-s + 5.00·12-s + 5.64·13-s − 9.64·14-s + 11.0·16-s + 4·17-s − 2.64·18-s − 19-s + 3.64·21-s − 14.9·22-s + 1.29·23-s − 7.93·24-s − 14.9·26-s + 27-s + 18.2·28-s − 6.93·29-s + 6·31-s − 13.2·32-s + 5.64·33-s − 10.5·34-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.577·3-s + 2.50·4-s − 1.08·6-s + 1.37·7-s − 2.80·8-s + 0.333·9-s + 1.70·11-s + 1.44·12-s + 1.56·13-s − 2.57·14-s + 2.75·16-s + 0.970·17-s − 0.623·18-s − 0.229·19-s + 0.795·21-s − 3.18·22-s + 0.269·23-s − 1.62·24-s − 2.92·26-s + 0.192·27-s + 3.44·28-s − 1.28·29-s + 1.07·31-s − 2.33·32-s + 0.982·33-s − 1.81·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.318990558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318990558\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 5.64T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 + 6.93T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 + 4.35T + 41T^{2} \) |
| 43 | \( 1 + 0.354T + 43T^{2} \) |
| 47 | \( 1 + 9.29T + 47T^{2} \) |
| 53 | \( 1 + 0.708T + 53T^{2} \) |
| 59 | \( 1 - 0.708T + 59T^{2} \) |
| 61 | \( 1 + 0.708T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 1.06T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−9.324897704878158269419186063575, −8.643139240401713534112512363744, −8.272003315255411304844981485799, −7.49076444965092625297836881843, −6.64773428201087790529392224422, −5.81299368671187477015046218556, −4.21442661820592900103616670130, −3.13866586407307833967892557544, −1.54330696656937962354340596612, −1.36556345711745183265884731997,
1.36556345711745183265884731997, 1.54330696656937962354340596612, 3.13866586407307833967892557544, 4.21442661820592900103616670130, 5.81299368671187477015046218556, 6.64773428201087790529392224422, 7.49076444965092625297836881843, 8.272003315255411304844981485799, 8.643139240401713534112512363744, 9.324897704878158269419186063575
