| L(s) = 1 | − 0.886·2-s + 9·3-s − 31.2·4-s − 7.97·6-s + 224.·7-s + 56.0·8-s + 81·9-s + 121·11-s − 280.·12-s + 1.06e3·13-s − 199.·14-s + 949.·16-s − 1.34e3·17-s − 71.8·18-s + 691.·19-s + 2.02e3·21-s − 107.·22-s + 3.39e3·23-s + 504.·24-s − 944.·26-s + 729·27-s − 7.01e3·28-s + 8.60e3·29-s − 320.·31-s − 2.63e3·32-s + 1.08e3·33-s + 1.19e3·34-s + ⋯ |
| L(s) = 1 | − 0.156·2-s + 0.577·3-s − 0.975·4-s − 0.0904·6-s + 1.73·7-s + 0.309·8-s + 0.333·9-s + 0.301·11-s − 0.563·12-s + 1.74·13-s − 0.271·14-s + 0.926·16-s − 1.13·17-s − 0.0522·18-s + 0.439·19-s + 1.00·21-s − 0.0472·22-s + 1.33·23-s + 0.178·24-s − 0.274·26-s + 0.192·27-s − 1.69·28-s + 1.89·29-s − 0.0599·31-s − 0.454·32-s + 0.174·33-s + 0.177·34-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
| \(L(3)\) |
\(\approx\) |
\(3.299564746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.299564746\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 0.886T + 32T^{2} \) |
| 7 | \( 1 - 224.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.06e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.34e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 691.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 320.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.42e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.45e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.10e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.88e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.77e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.13e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.15e5T + 8.58e9T^{2} \) |
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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
−9.023379865764342962666043178096, −8.702560491630807047413330712705, −8.129720827124642268803093278394, −7.12625554045203538088999916813, −5.86553830453701709960775299520, −4.71145252455652317400083014598, −4.29514817200029544740577060906, −3.07712473589331097144553039461, −1.56789029136306637902983738139, −0.952802378600297660443546801841,
0.952802378600297660443546801841, 1.56789029136306637902983738139, 3.07712473589331097144553039461, 4.29514817200029544740577060906, 4.71145252455652317400083014598, 5.86553830453701709960775299520, 7.12625554045203538088999916813, 8.129720827124642268803093278394, 8.702560491630807047413330712705, 9.023379865764342962666043178096
