| L(s) = 1 | + 0.758·2-s − 2.47·3-s − 1.42·4-s + 2.86·5-s − 1.88·6-s − 7-s − 2.59·8-s + 3.15·9-s + 2.17·10-s − 4.39·11-s + 3.53·12-s − 1.67·13-s − 0.758·14-s − 7.10·15-s + 0.877·16-s − 7.16·17-s + 2.39·18-s − 8.07·19-s − 4.08·20-s + 2.47·21-s − 3.33·22-s + 8.31·23-s + 6.44·24-s + 3.21·25-s − 1.26·26-s − 0.372·27-s + 1.42·28-s + ⋯ |
| L(s) = 1 | + 0.536·2-s − 1.43·3-s − 0.712·4-s + 1.28·5-s − 0.768·6-s − 0.377·7-s − 0.918·8-s + 1.05·9-s + 0.687·10-s − 1.32·11-s + 1.01·12-s − 0.464·13-s − 0.202·14-s − 1.83·15-s + 0.219·16-s − 1.73·17-s + 0.563·18-s − 1.85·19-s − 0.912·20-s + 0.541·21-s − 0.710·22-s + 1.73·23-s + 1.31·24-s + 0.643·25-s − 0.249·26-s − 0.0716·27-s + 0.269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 0.758T + 2T^{2} \) |
| 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 11 | \( 1 + 4.39T + 11T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 + 7.16T + 17T^{2} \) |
| 19 | \( 1 + 8.07T + 19T^{2} \) |
| 23 | \( 1 - 8.31T + 23T^{2} \) |
| 29 | \( 1 + 0.541T + 29T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 8.86T + 53T^{2} \) |
| 59 | \( 1 + 8.90T + 59T^{2} \) |
| 61 | \( 1 + 1.20T + 61T^{2} \) |
| 67 | \( 1 - 4.85T + 67T^{2} \) |
| 71 | \( 1 - 0.445T + 71T^{2} \) |
| 73 | \( 1 - 8.91T + 73T^{2} \) |
| 79 | \( 1 + 4.27T + 79T^{2} \) |
| 83 | \( 1 - 0.258T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 0.838T + 97T^{2} \) |
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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
−11.09683125026511700645241343271, −10.56014313078759522367429411914, −9.557961551164317400582486879214, −8.644618725383082520435787828477, −6.80328662047232501684579876133, −6.05302088808668228774456643685, −5.21786801721816166527664108536, −4.53977502990486978565358358401, −2.49895130475932552239254418407, 0,
2.49895130475932552239254418407, 4.53977502990486978565358358401, 5.21786801721816166527664108536, 6.05302088808668228774456643685, 6.80328662047232501684579876133, 8.644618725383082520435787828477, 9.557961551164317400582486879214, 10.56014313078759522367429411914, 11.09683125026511700645241343271
