| L(s) = 1 | + 3.54·2-s − 3·3-s + 4.59·4-s − 10.6·6-s + 21.2·7-s − 12.0·8-s + 9·9-s − 40.9·11-s − 13.7·12-s + 8.89·13-s + 75.4·14-s − 79.6·16-s + 7.68·17-s + 31.9·18-s + 40.7·19-s − 63.8·21-s − 145.·22-s + 80.7·23-s + 36.2·24-s + 31.5·26-s − 27·27-s + 97.6·28-s + 29·29-s + 82.1·31-s − 185.·32-s + 122.·33-s + 27.2·34-s + ⋯ |
| L(s) = 1 | + 1.25·2-s − 0.577·3-s + 0.574·4-s − 0.724·6-s + 1.14·7-s − 0.534·8-s + 0.333·9-s − 1.12·11-s − 0.331·12-s + 0.189·13-s + 1.44·14-s − 1.24·16-s + 0.109·17-s + 0.418·18-s + 0.492·19-s − 0.662·21-s − 1.40·22-s + 0.731·23-s + 0.308·24-s + 0.238·26-s − 0.192·27-s + 0.659·28-s + 0.185·29-s + 0.475·31-s − 1.02·32-s + 0.647·33-s + 0.137·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 3.54T + 8T^{2} \) |
| 7 | \( 1 - 21.2T + 343T^{2} \) |
| 11 | \( 1 + 40.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.89T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.68T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 80.7T + 1.21e4T^{2} \) |
| 31 | \( 1 - 82.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 53.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 17.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 23.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 399.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 388.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 399.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 92.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 979.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 62.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 163.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.26e3T + 9.12e5T^{2} \) |
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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
−8.215857464407133455691824508777, −7.39058772630860145599546221831, −6.52528864052710987657029857410, −5.57650305545904299353778142223, −5.07542339780777132000200655321, −4.58976933357116386980109937189, −3.51387975620622395815132305435, −2.58821909985484204324216621721, −1.39377174948631013603851612921, 0,
1.39377174948631013603851612921, 2.58821909985484204324216621721, 3.51387975620622395815132305435, 4.58976933357116386980109937189, 5.07542339780777132000200655321, 5.57650305545904299353778142223, 6.52528864052710987657029857410, 7.39058772630860145599546221831, 8.215857464407133455691824508777
