L(s) = 1 | − 3-s + 7-s + 9-s + 6·11-s − 2·13-s + 4·19-s − 21-s − 6·23-s − 5·25-s − 27-s − 6·29-s + 8·31-s − 6·33-s − 2·37-s + 2·39-s + 12·41-s + 4·43-s + 12·47-s + 49-s + 6·53-s − 4·57-s + 10·61-s + 63-s − 8·67-s + 6·69-s + 6·71-s − 10·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.917·19-s − 0.218·21-s − 1.25·23-s − 25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 1.04·33-s − 0.328·37-s + 0.320·39-s + 1.87·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.125·63-s − 0.977·67-s + 0.722·69-s + 0.712·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.548788080\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.548788080\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.583604842254255789237140171162, −9.000844289389290993593497586164, −7.82437442960794093330464733075, −7.19234324757373870654113009154, −6.18151745230772575654613193139, −5.60383328043951209454966346428, −4.37831328158321326146946665271, −3.81005665439346412696166766304, −2.20954050006078144234673698125, −0.985503276739119959400631494839,
0.985503276739119959400631494839, 2.20954050006078144234673698125, 3.81005665439346412696166766304, 4.37831328158321326146946665271, 5.60383328043951209454966346428, 6.18151745230772575654613193139, 7.19234324757373870654113009154, 7.82437442960794093330464733075, 9.000844289389290993593497586164, 9.583604842254255789237140171162