L(s) = 1 | + 5-s + 7-s + 11-s − 6·13-s − 2·17-s + 6·23-s + 25-s − 6·29-s + 2·31-s + 35-s + 10·37-s + 8·41-s + 8·43-s + 4·47-s + 49-s − 6·53-s + 55-s − 6·59-s − 8·61-s − 6·65-s − 14·67-s − 8·71-s − 2·73-s + 77-s + 16·79-s + 12·83-s − 2·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.301·11-s − 1.66·13-s − 0.485·17-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 0.169·35-s + 1.64·37-s + 1.24·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 0.134·55-s − 0.781·59-s − 1.02·61-s − 0.744·65-s − 1.71·67-s − 0.949·71-s − 0.234·73-s + 0.113·77-s + 1.80·79-s + 1.31·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.411713456\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.411713456\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−14.62325529132014, −13.89399629846242, −13.35132822514109, −12.94690323141380, −12.26861786667656, −12.03438643940053, −11.19666490448583, −10.88226514523395, −10.37388728784594, −9.568241608050571, −9.189064593076081, −9.050007693885288, −7.934983633648613, −7.558309093163323, −7.211721015500132, −6.345245700525977, −5.971505782549208, −5.238744015957115, −4.613294673478875, −4.378714334060888, −3.361304389963972, −2.615115194584999, −2.249990861709461, −1.366141314328262, −0.5467092855353353,
0.5467092855353353, 1.366141314328262, 2.249990861709461, 2.615115194584999, 3.361304389963972, 4.378714334060888, 4.613294673478875, 5.238744015957115, 5.971505782549208, 6.345245700525977, 7.211721015500132, 7.558309093163323, 7.934983633648613, 9.050007693885288, 9.189064593076081, 9.568241608050571, 10.37388728784594, 10.88226514523395, 11.19666490448583, 12.03438643940053, 12.26861786667656, 12.94690323141380, 13.35132822514109, 13.89399629846242, 14.62325529132014