L(s) = 1 | − 81·3-s − 625·5-s + 7.86e3·7-s + 6.56e3·9-s + 4.93e4·11-s + 2.42e4·13-s + 5.06e4·15-s + 2.68e5·17-s + 1.68e5·19-s − 6.36e5·21-s + 2.12e6·23-s + 3.90e5·25-s − 5.31e5·27-s + 3.89e5·29-s − 9.05e4·31-s − 3.99e6·33-s − 4.91e6·35-s − 3.31e6·37-s − 1.96e6·39-s + 2.32e7·41-s − 1.91e7·43-s − 4.10e6·45-s − 6.28e7·47-s + 2.14e7·49-s − 2.17e7·51-s − 1.80e5·53-s − 3.08e7·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.23·7-s + 0.333·9-s + 1.01·11-s + 0.235·13-s + 0.258·15-s + 0.778·17-s + 0.296·19-s − 0.714·21-s + 1.58·23-s + 0.200·25-s − 0.192·27-s + 0.102·29-s − 0.0176·31-s − 0.587·33-s − 0.553·35-s − 0.291·37-s − 0.135·39-s + 1.28·41-s − 0.852·43-s − 0.149·45-s − 1.87·47-s + 0.531·49-s − 0.449·51-s − 0.00313·53-s − 0.454·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(2.413684877\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.413684877\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 81T \) |
| 5 | \( 1 + 625T \) |
good | 7 | \( 1 - 7.86e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.93e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.42e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.68e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.89e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.05e4T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.31e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.91e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.28e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.80e5T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.84e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.53e5T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.39e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.28e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.39e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.28e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.12e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.07e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.70e9T + 7.60e17T^{2} \) |
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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
−10.82158894371469777128875876856, −9.552242086121137198211787286300, −8.493980629516209188301480561296, −7.55440567781701985667779926941, −6.56801355279333389309555021872, −5.31269352421356884224390810981, −4.50654095703370565632822371934, −3.32331781717757707483685028762, −1.59687161945713009187816229021, −0.817945586224562244081279131880,
0.817945586224562244081279131880, 1.59687161945713009187816229021, 3.32331781717757707483685028762, 4.50654095703370565632822371934, 5.31269352421356884224390810981, 6.56801355279333389309555021872, 7.55440567781701985667779926941, 8.493980629516209188301480561296, 9.552242086121137198211787286300, 10.82158894371469777128875876856