| L(s) = 1 | − 5.09·2-s − 9·3-s − 6.03·4-s + 45.8·6-s − 170.·7-s + 193.·8-s + 81·9-s + 121·11-s + 54.3·12-s − 364.·13-s + 867.·14-s − 794.·16-s − 1.30e3·17-s − 412.·18-s + 619.·19-s + 1.53e3·21-s − 616.·22-s + 922.·23-s − 1.74e3·24-s + 1.85e3·26-s − 729·27-s + 1.02e3·28-s − 3.43e3·29-s + 1.47e3·31-s − 2.15e3·32-s − 1.08e3·33-s + 6.66e3·34-s + ⋯ |
| L(s) = 1 | − 0.900·2-s − 0.577·3-s − 0.188·4-s + 0.520·6-s − 1.31·7-s + 1.07·8-s + 0.333·9-s + 0.301·11-s + 0.108·12-s − 0.597·13-s + 1.18·14-s − 0.775·16-s − 1.09·17-s − 0.300·18-s + 0.393·19-s + 0.758·21-s − 0.271·22-s + 0.363·23-s − 0.618·24-s + 0.538·26-s − 0.192·27-s + 0.247·28-s − 0.758·29-s + 0.276·31-s − 0.371·32-s − 0.174·33-s + 0.988·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 5.09T + 32T^{2} \) |
| 7 | \( 1 + 170.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 364.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.30e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 619.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 922.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.43e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.28e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.73e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.94e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.73e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.59e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.65e4T + 8.58e9T^{2} \) |
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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
−9.313100332261951787925125623696, −8.386043394504328413182111581175, −7.21557250373226455963961289837, −6.72729843428874407133115234102, −5.60654439586734230722976548303, −4.58189055893226697730537474967, −3.58448345401631718813891961014, −2.18400560898502746380573813445, −0.800759776808678939099559889674, 0,
0.800759776808678939099559889674, 2.18400560898502746380573813445, 3.58448345401631718813891961014, 4.58189055893226697730537474967, 5.60654439586734230722976548303, 6.72729843428874407133115234102, 7.21557250373226455963961289837, 8.386043394504328413182111581175, 9.313100332261951787925125623696
