| L(s) = 1 | + 0.784·2-s + 3-s − 1.38·4-s + 0.784·6-s − 1.51·7-s − 2.65·8-s + 9-s − 1.38·12-s − 0.466·13-s − 1.19·14-s + 0.688·16-s + 1.69·17-s + 0.784·18-s + 5.60·19-s − 1.51·21-s − 7.68·23-s − 2.65·24-s − 0.365·26-s + 27-s + 2.10·28-s + 2.66·29-s + 3.95·31-s + 5.84·32-s + 1.33·34-s − 1.38·36-s + 3.05·37-s + 4.39·38-s + ⋯ |
| L(s) = 1 | + 0.554·2-s + 0.577·3-s − 0.692·4-s + 0.320·6-s − 0.573·7-s − 0.938·8-s + 0.333·9-s − 0.399·12-s − 0.129·13-s − 0.318·14-s + 0.172·16-s + 0.411·17-s + 0.184·18-s + 1.28·19-s − 0.331·21-s − 1.60·23-s − 0.541·24-s − 0.0717·26-s + 0.192·27-s + 0.397·28-s + 0.494·29-s + 0.710·31-s + 1.03·32-s + 0.228·34-s − 0.230·36-s + 0.502·37-s + 0.713·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.784T + 2T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 13 | \( 1 + 0.466T + 13T^{2} \) |
| 17 | \( 1 - 1.69T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 + 1.35T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 2.55T + 47T^{2} \) |
| 53 | \( 1 + 8.01T + 53T^{2} \) |
| 59 | \( 1 - 7.54T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 6.89T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 2.26T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 1.47T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.53815482579268489332244183971, −6.48500491172917947475464446104, −6.07235215930504218773075539228, −5.10867006004168426021642479631, −4.65521853816268927041491384178, −3.62358773995583370416841456749, −3.35197577988467858086739606149, −2.46102800497477486076825067669, −1.25759640652744954083402971147, 0,
1.25759640652744954083402971147, 2.46102800497477486076825067669, 3.35197577988467858086739606149, 3.62358773995583370416841456749, 4.65521853816268927041491384178, 5.10867006004168426021642479631, 6.07235215930504218773075539228, 6.48500491172917947475464446104, 7.53815482579268489332244183971
