| L(s) = 1 | − 2-s + 4-s − 8-s + 4·11-s + 13-s + 16-s − 4·17-s − 4·22-s + 2·23-s − 26-s + 29-s + 4·31-s − 32-s + 4·34-s − 2·37-s − 4·43-s + 4·44-s − 2·46-s − 7·49-s + 52-s − 2·53-s − 58-s + 2·59-s + 10·61-s − 4·62-s + 64-s − 4·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.20·11-s + 0.277·13-s + 1/4·16-s − 0.970·17-s − 0.852·22-s + 0.417·23-s − 0.196·26-s + 0.185·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.328·37-s − 0.609·43-s + 0.603·44-s − 0.294·46-s − 49-s + 0.138·52-s − 0.274·53-s − 0.131·58-s + 0.260·59-s + 1.28·61-s − 0.508·62-s + 1/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.855714566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855714566\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.11960734248546, −12.83486963686004, −12.01385088707736, −11.78377776941858, −11.25775611447064, −10.96298386104741, −10.15272811601097, −10.01499475074539, −9.243068838663232, −8.978217676043025, −8.476333111359856, −8.108407703901339, −7.326253725861674, −6.940859285917531, −6.467475730632670, −6.094889283668354, −5.434975109119587, −4.605647987735531, −4.387651539729680, −3.459741131959167, −3.200831761967584, −2.273686121599510, −1.809223814156391, −1.122340715883706, −0.4854846337150848,
0.4854846337150848, 1.122340715883706, 1.809223814156391, 2.273686121599510, 3.200831761967584, 3.459741131959167, 4.387651539729680, 4.605647987735531, 5.434975109119587, 6.094889283668354, 6.467475730632670, 6.940859285917531, 7.326253725861674, 8.108407703901339, 8.476333111359856, 8.978217676043025, 9.243068838663232, 10.01499475074539, 10.15272811601097, 10.96298386104741, 11.25775611447064, 11.78377776941858, 12.01385088707736, 12.83486963686004, 13.11960734248546
