L(s) = 1 | − 2·2-s − 28·4-s + 132·7-s + 120·8-s − 472·11-s + 686·13-s − 264·14-s + 656·16-s − 1.56e3·17-s − 2.18e3·19-s + 944·22-s + 264·23-s − 1.37e3·26-s − 3.69e3·28-s − 170·29-s + 7.27e3·31-s − 5.15e3·32-s + 3.12e3·34-s + 142·37-s + 4.36e3·38-s + 1.61e4·41-s + 1.03e4·43-s + 1.32e4·44-s − 528·46-s + 1.85e4·47-s + 617·49-s − 1.92e4·52-s + ⋯ |
L(s) = 1 | − 0.353·2-s − 7/8·4-s + 1.01·7-s + 0.662·8-s − 1.17·11-s + 1.12·13-s − 0.359·14-s + 0.640·16-s − 1.31·17-s − 1.38·19-s + 0.415·22-s + 0.104·23-s − 0.398·26-s − 0.890·28-s − 0.0375·29-s + 1.35·31-s − 0.889·32-s + 0.463·34-s + 0.0170·37-s + 0.489·38-s + 1.50·41-s + 0.850·43-s + 1.02·44-s − 0.0367·46-s + 1.22·47-s + 0.0367·49-s − 0.985·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.257405093\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.257405093\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + p T + p^{5} T^{2} \) |
| 7 | \( 1 - 132 T + p^{5} T^{2} \) |
| 11 | \( 1 + 472 T + p^{5} T^{2} \) |
| 13 | \( 1 - 686 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1562 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2180 T + p^{5} T^{2} \) |
| 23 | \( 1 - 264 T + p^{5} T^{2} \) |
| 29 | \( 1 + 170 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7272 T + p^{5} T^{2} \) |
| 37 | \( 1 - 142 T + p^{5} T^{2} \) |
| 41 | \( 1 - 16198 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10316 T + p^{5} T^{2} \) |
| 47 | \( 1 - 18568 T + p^{5} T^{2} \) |
| 53 | \( 1 - 21514 T + p^{5} T^{2} \) |
| 59 | \( 1 + 34600 T + p^{5} T^{2} \) |
| 61 | \( 1 + 35738 T + p^{5} T^{2} \) |
| 67 | \( 1 - 5772 T + p^{5} T^{2} \) |
| 71 | \( 1 - 69088 T + p^{5} T^{2} \) |
| 73 | \( 1 - 70526 T + p^{5} T^{2} \) |
| 79 | \( 1 - 47640 T + p^{5} T^{2} \) |
| 83 | \( 1 - 74004 T + p^{5} T^{2} \) |
| 89 | \( 1 - 90030 T + p^{5} T^{2} \) |
| 97 | \( 1 - 33502 T + p^{5} T^{2} \) |
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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
−10.87990883679588545063734019119, −10.71019666781469943032338107378, −9.203573229402758379945278982473, −8.419337515910641690074033186641, −7.77447518440058863391877949415, −6.19219048924158129641597311644, −4.90851025181864403280587021437, −4.12331881089028883278302644378, −2.23988110029223652967370459876, −0.72031908795903763908152315873,
0.72031908795903763908152315873, 2.23988110029223652967370459876, 4.12331881089028883278302644378, 4.90851025181864403280587021437, 6.19219048924158129641597311644, 7.77447518440058863391877949415, 8.419337515910641690074033186641, 9.203573229402758379945278982473, 10.71019666781469943032338107378, 10.87990883679588545063734019119