| L(s) = 1 | − 38.2·2-s − 81·3-s + 948.·4-s − 2.36e3·5-s + 3.09e3·6-s + 4.56e3·7-s − 1.66e4·8-s + 6.56e3·9-s + 9.03e4·10-s − 4.82e4·11-s − 7.68e4·12-s + 1.91e5·13-s − 1.74e5·14-s + 1.91e5·15-s + 1.51e5·16-s + 3.38e5·17-s − 2.50e5·18-s + 5.24e5·19-s − 2.24e6·20-s − 3.69e5·21-s + 1.84e6·22-s − 1.14e6·23-s + 1.35e6·24-s + 3.63e6·25-s − 7.30e6·26-s − 5.31e5·27-s + 4.32e6·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.577·3-s + 1.85·4-s − 1.69·5-s + 0.975·6-s + 0.718·7-s − 1.43·8-s + 0.333·9-s + 2.85·10-s − 0.993·11-s − 1.06·12-s + 1.85·13-s − 1.21·14-s + 0.976·15-s + 0.578·16-s + 0.981·17-s − 0.562·18-s + 0.923·19-s − 3.13·20-s − 0.414·21-s + 1.67·22-s − 0.853·23-s + 0.830·24-s + 1.86·25-s − 3.13·26-s − 0.192·27-s + 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5849982144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5849982144\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 + 38.2T + 512T^{2} \) |
| 5 | \( 1 + 2.36e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.56e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.82e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.91e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.38e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.24e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.14e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.85e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.88e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.22e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.14e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.58e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.12e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.33e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.55e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.33e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.02e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.90e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.11e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.30e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.66e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.50e8T + 7.60e17T^{2} \) |
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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
−10.91829734733904979937644358187, −10.20706889978774782312228003083, −8.690964235133468992763103718992, −7.965475275911062953931117875941, −7.54453085507495987715186168367, −6.13783790170146902614330306986, −4.56795141801899934493984337286, −3.16129821677946523038976385323, −1.28269797371452426955810623926, −0.58851901178897894355165023411,
0.58851901178897894355165023411, 1.28269797371452426955810623926, 3.16129821677946523038976385323, 4.56795141801899934493984337286, 6.13783790170146902614330306986, 7.54453085507495987715186168367, 7.965475275911062953931117875941, 8.690964235133468992763103718992, 10.20706889978774782312228003083, 10.91829734733904979937644358187
