| L(s) = 1 | − 5-s + 11-s − 3·17-s + 3·19-s − 2·23-s + 25-s + 4·29-s − 6·31-s − 3·37-s − 7·41-s − 7·43-s − 9·47-s − 7·49-s − 55-s − 6·59-s + 6·61-s − 7·67-s + 10·71-s − 4·73-s + 6·79-s − 2·83-s + 3·85-s + 18·89-s − 3·95-s + 7·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s − 0.727·17-s + 0.688·19-s − 0.417·23-s + 1/5·25-s + 0.742·29-s − 1.07·31-s − 0.493·37-s − 1.09·41-s − 1.06·43-s − 1.31·47-s − 49-s − 0.134·55-s − 0.781·59-s + 0.768·61-s − 0.855·67-s + 1.18·71-s − 0.468·73-s + 0.675·79-s − 0.219·83-s + 0.325·85-s + 1.90·89-s − 0.307·95-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8880932151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8880932151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.60688808894446, −12.04432297793970, −11.61722287774278, −11.39015767166924, −10.82128027372667, −10.23444659097493, −9.980856553604287, −9.294033207628340, −8.973235831426482, −8.426768488331510, −8.003637904552549, −7.562817215372260, −6.963872347794803, −6.555772617982715, −6.216180568818173, −5.429068008025492, −4.967127756513823, −4.632342225847897, −3.937857066294086, −3.330674942116582, −3.199863300620039, −2.204745819732598, −1.784906655181050, −1.122965530331332, −0.2562309020599558,
0.2562309020599558, 1.122965530331332, 1.784906655181050, 2.204745819732598, 3.199863300620039, 3.330674942116582, 3.937857066294086, 4.632342225847897, 4.967127756513823, 5.429068008025492, 6.216180568818173, 6.555772617982715, 6.963872347794803, 7.562817215372260, 8.003637904552549, 8.426768488331510, 8.973235831426482, 9.294033207628340, 9.980856553604287, 10.23444659097493, 10.82128027372667, 11.39015767166924, 11.61722287774278, 12.04432297793970, 12.60688808894446
