| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 25·5-s + 36·6-s − 177.·7-s − 64·8-s + 81·9-s − 100·10-s − 464.·11-s − 144·12-s + 535.·13-s + 709.·14-s − 225·15-s + 256·16-s + 173.·17-s − 324·18-s + 361·19-s + 400·20-s + 1.59e3·21-s + 1.85e3·22-s + 2.05e3·23-s + 576·24-s + 625·25-s − 2.14e3·26-s − 729·27-s − 2.83e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.36·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.15·11-s − 0.288·12-s + 0.878·13-s + 0.967·14-s − 0.258·15-s + 0.250·16-s + 0.145·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.789·21-s + 0.818·22-s + 0.808·23-s + 0.204·24-s + 0.200·25-s − 0.621·26-s − 0.192·27-s − 0.684·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 5 | \( 1 - 25T \) |
| 19 | \( 1 - 361T \) |
| good | 7 | \( 1 + 177.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 464.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 535.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 173.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.05e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.40e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.70e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 731.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.41e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.42e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.79e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.63e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.97e4T + 8.58e9T^{2} \) |
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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
−9.583915497322751019796514302348, −8.837603862075191261124988078627, −7.69211509486292194487593902968, −6.73375260143736089650205127208, −6.03754677328473983322164165738, −5.14266377876641583340562290282, −3.52125773833281939904756082588, −2.53094550363241591353237435252, −1.04917506594246080307402499108, 0,
1.04917506594246080307402499108, 2.53094550363241591353237435252, 3.52125773833281939904756082588, 5.14266377876641583340562290282, 6.03754677328473983322164165738, 6.73375260143736089650205127208, 7.69211509486292194487593902968, 8.837603862075191261124988078627, 9.583915497322751019796514302348
