| L(s) = 1 | − 1.24i·2-s − 1.51i·3-s + 0.439·4-s + (−1.95 + 1.09i)5-s − 1.89·6-s + 0.568i·7-s − 3.04i·8-s + 0.710·9-s + (1.36 + 2.43i)10-s + 3.18·11-s − 0.664i·12-s − 2.28i·13-s + 0.710·14-s + (1.65 + 2.95i)15-s − 2.92·16-s + 2.21i·17-s + ⋯ |
| L(s) = 1 | − 0.883i·2-s − 0.873i·3-s + 0.219·4-s + (−0.872 + 0.489i)5-s − 0.771·6-s + 0.214i·7-s − 1.07i·8-s + 0.236·9-s + (0.432 + 0.770i)10-s + 0.961·11-s − 0.191i·12-s − 0.632i·13-s + 0.189·14-s + (0.427 + 0.761i)15-s − 0.732·16-s + 0.537i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.793823396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.793823396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.95 - 1.09i)T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.24iT - 2T^{2} \) |
| 3 | \( 1 + 1.51iT - 3T^{2} \) |
| 7 | \( 1 - 0.568iT - 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 + 2.28iT - 13T^{2} \) |
| 17 | \( 1 - 2.21iT - 17T^{2} \) |
| 23 | \( 1 + 0.0101iT - 23T^{2} \) |
| 29 | \( 1 - 1.14T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 8.97iT - 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.91iT - 43T^{2} \) |
| 47 | \( 1 + 12.5iT - 47T^{2} \) |
| 53 | \( 1 - 7.43iT - 53T^{2} \) |
| 59 | \( 1 - 8.46T + 59T^{2} \) |
| 61 | \( 1 + 7.36T + 61T^{2} \) |
| 67 | \( 1 + 3.51iT - 67T^{2} \) |
| 71 | \( 1 + 4.56T + 71T^{2} \) |
| 73 | \( 1 + 9.66iT - 73T^{2} \) |
| 79 | \( 1 + 7.97T + 79T^{2} \) |
| 83 | \( 1 + 13.7iT - 83T^{2} \) |
| 89 | \( 1 + 3.41T + 89T^{2} \) |
| 97 | \( 1 + 3.01iT - 97T^{2} \) |
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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
−8.906996396714420022897816112520, −8.055804484497463511561927580193, −7.29136900480113969294625669632, −6.69243838113122143597278026445, −6.03410023464791260395808284150, −4.45519508948041002382754228920, −3.66521352804345538843275258265, −2.79589049997958552072976881604, −1.77560749882316357066124097119, −0.72839586671991602492562013115,
1.36993092933857482439690655228, 3.01826979867372514038328051566, 4.12827599054599930508063984378, 4.62664470861725549949502151897, 5.48353764611134585559188723279, 6.78555312203208530959922522635, 6.94651338514801838107277671181, 8.103942453360200085763633081179, 8.634847369920906622664500687013, 9.474079411402694815396256141721
