| L(s) = 1 | − 18.5·5-s + 9.32·7-s − 39.7·11-s − 32.9·13-s − 90.5·17-s − 72.5·19-s − 45.3·23-s + 218.·25-s − 143.·29-s − 90.4·31-s − 172.·35-s + 1.77·37-s + 195.·41-s − 407.·43-s − 278.·47-s − 256.·49-s − 241.·53-s + 736.·55-s − 149.·59-s + 508.·61-s + 611.·65-s − 950.·67-s + 803.·71-s − 449.·73-s − 370.·77-s + 157.·79-s − 175.·83-s + ⋯ |
| L(s) = 1 | − 1.65·5-s + 0.503·7-s − 1.08·11-s − 0.703·13-s − 1.29·17-s − 0.876·19-s − 0.411·23-s + 1.75·25-s − 0.918·29-s − 0.524·31-s − 0.835·35-s + 0.00790·37-s + 0.745·41-s − 1.44·43-s − 0.864·47-s − 0.746·49-s − 0.625·53-s + 1.80·55-s − 0.330·59-s + 1.06·61-s + 1.16·65-s − 1.73·67-s + 1.34·71-s − 0.721·73-s − 0.548·77-s + 0.223·79-s − 0.231·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09626697101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09626697101\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 18.5T + 125T^{2} \) |
| 7 | \( 1 - 9.32T + 343T^{2} \) |
| 11 | \( 1 + 39.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 72.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 143.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 90.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 1.77T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 278.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 241.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 149.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 508.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 950.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 803.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 449.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 157.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 175.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 127.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 158.T + 9.12e5T^{2} \) |
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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
−8.360462261953078027123394934394, −7.967058542861205833713338658914, −7.29291578644939668147593265495, −6.51967495582573731412024093307, −5.24844790082389519866567378358, −4.58144922842678958115620180271, −3.92005224104757685305385291567, −2.87558580978803821072781452774, −1.87086694723699650879598079078, −0.13092154094516947552086111889,
0.13092154094516947552086111889, 1.87086694723699650879598079078, 2.87558580978803821072781452774, 3.92005224104757685305385291567, 4.58144922842678958115620180271, 5.24844790082389519866567378358, 6.51967495582573731412024093307, 7.29291578644939668147593265495, 7.967058542861205833713338658914, 8.360462261953078027123394934394
