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
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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
−8.348752472078381031637939738785, −6.82876636275730201103002915091, −6.52380563790650395256302617522, −5.98167716574475503512952756192, −5.49052749462300012283644548757, −4.32751388823861433388128374344, −3.60073927284605671710273959125, −1.84375225055718099348339490184, −1.15440612194477680239692256459, 0,
1.15440612194477680239692256459, 1.84375225055718099348339490184, 3.60073927284605671710273959125, 4.32751388823861433388128374344, 5.49052749462300012283644548757, 5.98167716574475503512952756192, 6.52380563790650395256302617522, 6.82876636275730201103002915091, 8.348752472078381031637939738785