L(s) = 1 | − 43.0·2-s + 1.34e3·4-s + 1.33e3·5-s − 743.·7-s − 3.58e4·8-s − 5.73e4·10-s + 6.08e4·11-s − 1.04e5·13-s + 3.20e4·14-s + 8.58e5·16-s + 3.44e5·17-s − 6.26e5·19-s + 1.79e6·20-s − 2.62e6·22-s − 1.13e6·23-s − 1.81e5·25-s + 4.48e6·26-s − 1.00e6·28-s − 3.51e6·29-s − 6.97e6·31-s − 1.86e7·32-s − 1.48e7·34-s − 9.89e5·35-s + 7.60e6·37-s + 2.70e7·38-s − 4.77e7·40-s − 3.04e7·41-s + ⋯ |
L(s) = 1 | − 1.90·2-s + 2.62·4-s + 0.952·5-s − 0.117·7-s − 3.09·8-s − 1.81·10-s + 1.25·11-s − 1.01·13-s + 0.222·14-s + 3.27·16-s + 1.00·17-s − 1.10·19-s + 2.50·20-s − 2.38·22-s − 0.842·23-s − 0.0929·25-s + 1.92·26-s − 0.307·28-s − 0.922·29-s − 1.35·31-s − 3.13·32-s − 1.90·34-s − 0.111·35-s + 0.666·37-s + 2.10·38-s − 2.95·40-s − 1.68·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.8545666057\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8545666057\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 + 3.41e6T \) |
good | 2 | \( 1 + 43.0T + 512T^{2} \) |
| 5 | \( 1 - 1.33e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 743.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.08e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.04e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.44e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.26e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.13e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.51e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.60e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.04e7T + 3.27e14T^{2} \) |
| 47 | \( 1 + 3.57e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.41e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.25e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.62e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.76e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 7.75e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.87e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.61e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.53e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.02e9T + 7.60e17T^{2} \) |
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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
−9.857754976806218632822506769947, −9.058416139486146050299110186216, −8.197041089593507225688246751098, −7.17090005517898504931079618238, −6.44303223482752625824587838237, −5.53533241615143394963442539313, −3.60311775112092644609198183048, −2.15014046606681922743458737849, −1.71590774198702733165711745833, −0.50942267096239345314348907280,
0.50942267096239345314348907280, 1.71590774198702733165711745833, 2.15014046606681922743458737849, 3.60311775112092644609198183048, 5.53533241615143394963442539313, 6.44303223482752625824587838237, 7.17090005517898504931079618238, 8.197041089593507225688246751098, 9.058416139486146050299110186216, 9.857754976806218632822506769947