| L(s) = 1 | + 8.82·7-s + 9·9-s + 17.3·11-s + 33.9·17-s − 19·19-s + 30·23-s − 31.1·43-s − 11.5·47-s + 28.8·49-s − 108.·61-s + 79.4·63-s − 137.·73-s + 153.·77-s + 81·81-s − 90·83-s + 156.·99-s − 102·101-s + 299.·119-s + ⋯ |
| L(s) = 1 | + 1.26·7-s + 9-s + 1.57·11-s + 1.99·17-s − 19-s + 1.30·23-s − 0.725·43-s − 0.246·47-s + 0.589·49-s − 1.77·61-s + 1.26·63-s − 1.87·73-s + 1.99·77-s + 81-s − 1.08·83-s + 1.57·99-s − 1.00·101-s + 2.51·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.274551675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.274551675\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 9T^{2} \) |
| 7 | \( 1 - 8.82T + 49T^{2} \) |
| 11 | \( 1 - 17.3T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 33.9T + 289T^{2} \) |
| 23 | \( 1 - 30T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 31.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 90T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.010336556512219187556098014729, −8.237067162843546821653440422386, −7.45422462746594013909818768646, −6.78345162429324655950345296517, −5.81465927517559651812578099604, −4.82409167807978054393238039799, −4.19422490477833243840068315924, −3.22455640702439263994379769461, −1.63934699019100711331845484937, −1.17391223682618638231420277467,
1.17391223682618638231420277467, 1.63934699019100711331845484937, 3.22455640702439263994379769461, 4.19422490477833243840068315924, 4.82409167807978054393238039799, 5.81465927517559651812578099604, 6.78345162429324655950345296517, 7.45422462746594013909818768646, 8.237067162843546821653440422386, 9.010336556512219187556098014729
