| L(s) = 1 | − 0.644·2-s − 7.58·4-s − 5·5-s + 10.0·8-s + 3.22·10-s + 47.7·11-s − 57.2·13-s + 54.1·16-s − 36.9·17-s − 30.7·19-s + 37.9·20-s − 30.7·22-s − 53.1·23-s + 25·25-s + 36.8·26-s + 195.·29-s − 257.·31-s − 115.·32-s + 23.8·34-s + 346.·37-s + 19.8·38-s − 50.2·40-s + 267.·41-s − 176.·43-s − 361.·44-s + 34.2·46-s − 311.·47-s + ⋯ |
| L(s) = 1 | − 0.227·2-s − 0.948·4-s − 0.447·5-s + 0.443·8-s + 0.101·10-s + 1.30·11-s − 1.22·13-s + 0.846·16-s − 0.527·17-s − 0.371·19-s + 0.423·20-s − 0.298·22-s − 0.481·23-s + 0.200·25-s + 0.278·26-s + 1.25·29-s − 1.49·31-s − 0.637·32-s + 0.120·34-s + 1.53·37-s + 0.0846·38-s − 0.198·40-s + 1.01·41-s − 0.627·43-s − 1.23·44-s + 0.109·46-s − 0.967·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.644T + 8T^{2} \) |
| 11 | \( 1 - 47.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 57.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 36.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 30.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 195.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 257.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 267.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 311.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 98.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 82.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 654.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 779.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 829.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 769.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 613.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 457.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−8.416967073773433515706622400379, −7.63884260737345136390926414218, −6.87300748463116316141211816636, −5.96944177626039225073422574190, −4.87312129884779533239193993492, −4.30462885022467551900177280186, −3.55914854678792437858934322195, −2.26086568436114719232449294986, −0.998178678086866782112530470345, 0,
0.998178678086866782112530470345, 2.26086568436114719232449294986, 3.55914854678792437858934322195, 4.30462885022467551900177280186, 4.87312129884779533239193993492, 5.96944177626039225073422574190, 6.87300748463116316141211816636, 7.63884260737345136390926414218, 8.416967073773433515706622400379
