| L(s) = 1 | + 1.01·2-s − 2.79·3-s − 0.979·4-s + 5-s − 2.82·6-s − 1.25·7-s − 3.00·8-s + 4.82·9-s + 1.01·10-s − 3.20·11-s + 2.74·12-s − 2.07·13-s − 1.26·14-s − 2.79·15-s − 1.08·16-s − 5.18·17-s + 4.86·18-s − 0.979·20-s + 3.51·21-s − 3.24·22-s + 9.33·23-s + 8.41·24-s + 25-s − 2.09·26-s − 5.09·27-s + 1.23·28-s − 8.56·29-s + ⋯ |
| L(s) = 1 | + 0.714·2-s − 1.61·3-s − 0.489·4-s + 0.447·5-s − 1.15·6-s − 0.474·7-s − 1.06·8-s + 1.60·9-s + 0.319·10-s − 0.967·11-s + 0.790·12-s − 0.575·13-s − 0.338·14-s − 0.722·15-s − 0.270·16-s − 1.25·17-s + 1.14·18-s − 0.219·20-s + 0.766·21-s − 0.691·22-s + 1.94·23-s + 1.71·24-s + 0.200·25-s − 0.410·26-s − 0.980·27-s + 0.232·28-s − 1.58·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6420993713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6420993713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.01T + 2T^{2} \) |
| 3 | \( 1 + 2.79T + 3T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 + 3.20T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 + 5.18T + 17T^{2} \) |
| 23 | \( 1 - 9.33T + 23T^{2} \) |
| 29 | \( 1 + 8.56T + 29T^{2} \) |
| 31 | \( 1 + 3.10T + 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 41 | \( 1 + 4.42T + 41T^{2} \) |
| 43 | \( 1 - 7.55T + 43T^{2} \) |
| 47 | \( 1 - 5.90T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 2.01T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 0.135T + 71T^{2} \) |
| 73 | \( 1 - 6.62T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 - 1.79T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 4.67T + 97T^{2} \) |
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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
−9.382089367511351199500413248806, −8.696755946482799243805973590252, −7.20955233403313420711918327926, −6.71581064412393760280835195166, −5.68750578743878645508139369616, −5.29007306074454467497198126383, −4.70440391993889226504042960018, −3.62010676553496483359050554193, −2.35375125202479289670552462002, −0.50941828176835194343435086140,
0.50941828176835194343435086140, 2.35375125202479289670552462002, 3.62010676553496483359050554193, 4.70440391993889226504042960018, 5.29007306074454467497198126383, 5.68750578743878645508139369616, 6.71581064412393760280835195166, 7.20955233403313420711918327926, 8.696755946482799243805973590252, 9.382089367511351199500413248806
