| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s − 11-s + 6·13-s + 16-s − 5·17-s + 7·19-s + 2·20-s − 22-s − 4·23-s − 25-s + 6·26-s + 4·29-s + 6·31-s + 32-s − 5·34-s + 2·37-s + 7·38-s + 2·40-s + 3·41-s − 43-s − 44-s − 4·46-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.301·11-s + 1.66·13-s + 1/4·16-s − 1.21·17-s + 1.60·19-s + 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s + 1.17·26-s + 0.742·29-s + 1.07·31-s + 0.176·32-s − 0.857·34-s + 0.328·37-s + 1.13·38-s + 0.316·40-s + 0.468·41-s − 0.152·43-s − 0.150·44-s − 0.589·46-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.476587216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.476587216\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−7.939634904786298294281535153848, −6.81857948715714490646689235780, −6.34513750707259648884330030169, −5.79006764751200754802324926390, −5.11726338036088083512560037454, −4.31122598605942961207910830347, −3.54214053252758259126083146499, −2.72888017782480302102905723674, −1.90223486181273855752300864459, −0.992490936632162384684264681047,
0.992490936632162384684264681047, 1.90223486181273855752300864459, 2.72888017782480302102905723674, 3.54214053252758259126083146499, 4.31122598605942961207910830347, 5.11726338036088083512560037454, 5.79006764751200754802324926390, 6.34513750707259648884330030169, 6.81857948715714490646689235780, 7.939634904786298294281535153848
