| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 4·11-s − 6·13-s − 14-s + 16-s + 4·17-s − 4·22-s + 4·23-s − 6·26-s − 28-s + 3·29-s + 7·31-s + 32-s + 4·34-s + 37-s − 7·41-s + 10·43-s − 4·44-s + 4·46-s − 13·47-s + 49-s − 6·52-s + 6·53-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.852·22-s + 0.834·23-s − 1.17·26-s − 0.188·28-s + 0.557·29-s + 1.25·31-s + 0.176·32-s + 0.685·34-s + 0.164·37-s − 1.09·41-s + 1.52·43-s − 0.603·44-s + 0.589·46-s − 1.89·47-s + 1/7·49-s − 0.832·52-s + 0.824·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.499501741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.499501741\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.69602961001513661056380519097, −6.96533992681346908069894123810, −6.31203248808389235949212976512, −5.44453454674948555622222880890, −4.96525116316353916486634790192, −4.41996149418409880942132947708, −3.17910447470567868575025103846, −2.87995165879104634451857584655, −2.01118430099902388274022232870, −0.65060005861330759911928049647,
0.65060005861330759911928049647, 2.01118430099902388274022232870, 2.87995165879104634451857584655, 3.17910447470567868575025103846, 4.41996149418409880942132947708, 4.96525116316353916486634790192, 5.44453454674948555622222880890, 6.31203248808389235949212976512, 6.96533992681346908069894123810, 7.69602961001513661056380519097
