| L(s) = 1 | + 134.·3-s − 1.08e3·5-s − 2.40e3·7-s − 1.70e3·9-s + 8.46e4·11-s − 1.13e5·13-s − 1.46e5·15-s + 4.62e5·17-s + 8.62e5·19-s − 3.21e5·21-s − 2.32e6·23-s − 7.66e5·25-s − 2.86e6·27-s − 3.09e6·29-s − 7.49e6·31-s + 1.13e7·33-s + 2.61e6·35-s + 7.68e6·37-s − 1.51e7·39-s − 2.07e7·41-s − 3.51e7·43-s + 1.86e6·45-s + 8.14e6·47-s + 5.76e6·49-s + 6.20e7·51-s + 5.77e7·53-s − 9.22e7·55-s + ⋯ |
| L(s) = 1 | + 0.955·3-s − 0.779·5-s − 0.377·7-s − 0.0868·9-s + 1.74·11-s − 1.09·13-s − 0.744·15-s + 1.34·17-s + 1.51·19-s − 0.361·21-s − 1.73·23-s − 0.392·25-s − 1.03·27-s − 0.812·29-s − 1.45·31-s + 1.66·33-s + 0.294·35-s + 0.674·37-s − 1.05·39-s − 1.14·41-s − 1.56·43-s + 0.0677·45-s + 0.243·47-s + 0.142·49-s + 1.28·51-s + 1.00·53-s − 1.35·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 3 | \( 1 - 134.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.08e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 8.46e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.13e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.62e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.62e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.32e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.09e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.49e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.68e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.07e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.51e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 8.14e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.77e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.77e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.84e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.10e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.24e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.62e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.96e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.18e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.50e7T + 7.60e17T^{2} \) |
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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
−11.83018583796054224316139370027, −9.869308321892083128303585578389, −9.251345612431190473312110108192, −7.979157824613526306357435753845, −7.24047002451872270653122063240, −5.67515924200277647867286895261, −3.91145954367286356086998226748, −3.25870168234043045645621683375, −1.66317198419316108701481729005, 0,
1.66317198419316108701481729005, 3.25870168234043045645621683375, 3.91145954367286356086998226748, 5.67515924200277647867286895261, 7.24047002451872270653122063240, 7.979157824613526306357435753845, 9.251345612431190473312110108192, 9.869308321892083128303585578389, 11.83018583796054224316139370027
