| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 4·11-s + 2·13-s + 14-s + 16-s + 2·17-s + 4·19-s + 4·22-s − 8·23-s + 2·26-s + 28-s + 2·29-s + 32-s + 2·34-s − 6·37-s + 4·38-s + 6·41-s + 4·43-s + 4·44-s − 8·46-s + 49-s + 2·52-s − 10·53-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 + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.852·22-s − 1.66·23-s + 0.392·26-s + 0.188·28-s + 0.371·29-s + 0.176·32-s + 0.342·34-s − 0.986·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s + 1/7·49-s + 0.277·52-s − 1.37·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.560276599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.560276599\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.572361691247425818962009402284, −7.87057848456857363225229546010, −7.09304870672129430507335279038, −6.24664410799738980273854798490, −5.70967592536431225977171521773, −4.75782304131629250034613305389, −3.93839119387133171902822326777, −3.33286092856857532309906421674, −2.07302940285776049892947889691, −1.12260455198864207776618149898,
1.12260455198864207776618149898, 2.07302940285776049892947889691, 3.33286092856857532309906421674, 3.93839119387133171902822326777, 4.75782304131629250034613305389, 5.70967592536431225977171521773, 6.24664410799738980273854798490, 7.09304870672129430507335279038, 7.87057848456857363225229546010, 8.572361691247425818962009402284
