| L(s) = 1 | + 10.0·3-s − 13.8·5-s + 18.1·7-s + 73.9·9-s − 46.2·11-s + 65.8·13-s − 139.·15-s − 19.0·19-s + 181.·21-s + 194.·23-s + 66.7·25-s + 471.·27-s − 118.·29-s + 4.17·31-s − 465.·33-s − 250.·35-s + 270.·37-s + 661.·39-s − 31.6·41-s + 9.51·43-s − 1.02e3·45-s − 471.·47-s − 15.0·49-s − 96.9·53-s + 641.·55-s − 191.·57-s + 854.·59-s + ⋯ |
| L(s) = 1 | + 1.93·3-s − 1.23·5-s + 0.977·7-s + 2.73·9-s − 1.26·11-s + 1.40·13-s − 2.39·15-s − 0.230·19-s + 1.89·21-s + 1.76·23-s + 0.534·25-s + 3.35·27-s − 0.759·29-s + 0.0241·31-s − 2.45·33-s − 1.21·35-s + 1.20·37-s + 2.71·39-s − 0.120·41-s + 0.0337·43-s − 3.39·45-s − 1.46·47-s − 0.0439·49-s − 0.251·53-s + 1.57·55-s − 0.444·57-s + 1.88·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.784534486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.784534486\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 10.0T + 27T^{2} \) |
| 5 | \( 1 + 13.8T + 125T^{2} \) |
| 7 | \( 1 - 18.1T + 343T^{2} \) |
| 11 | \( 1 + 46.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.8T + 2.19e3T^{2} \) |
| 19 | \( 1 + 19.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 194.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 118.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 4.17T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 31.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 9.51T + 7.95e4T^{2} \) |
| 47 | \( 1 + 471.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 96.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 854.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 171.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 8.33T + 3.00e5T^{2} \) |
| 71 | \( 1 - 51.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 946.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 16.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.416757269301908003263056328342, −8.016938948616497969579043237209, −7.58044862562865052099904733933, −6.72207514123250771606061827421, −5.15813553728279667968407239209, −4.37615116558415049173489949942, −3.61102814054325026166461096655, −2.97944770159277582777895304424, −1.97615938876352291837755177285, −0.928316512684452458514944575203,
0.928316512684452458514944575203, 1.97615938876352291837755177285, 2.97944770159277582777895304424, 3.61102814054325026166461096655, 4.37615116558415049173489949942, 5.15813553728279667968407239209, 6.72207514123250771606061827421, 7.58044862562865052099904733933, 8.016938948616497969579043237209, 8.416757269301908003263056328342
