| L(s) = 1 | − 2-s − 2.81·3-s + 4-s + 2.81·6-s − 0.647·7-s − 8-s + 4.91·9-s − 3.05·11-s − 2.81·12-s + 0.606·13-s + 0.647·14-s + 16-s + 2.27·17-s − 4.91·18-s + 3.91·19-s + 1.82·21-s + 3.05·22-s − 8.36·23-s + 2.81·24-s − 0.606·26-s − 5.37·27-s − 0.647·28-s + 29-s + 4.40·31-s − 32-s + 8.59·33-s − 2.27·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.62·3-s + 0.5·4-s + 1.14·6-s − 0.244·7-s − 0.353·8-s + 1.63·9-s − 0.921·11-s − 0.811·12-s + 0.168·13-s + 0.173·14-s + 0.250·16-s + 0.551·17-s − 1.15·18-s + 0.898·19-s + 0.397·21-s + 0.651·22-s − 1.74·23-s + 0.574·24-s − 0.118·26-s − 1.03·27-s − 0.122·28-s + 0.185·29-s + 0.791·31-s − 0.176·32-s + 1.49·33-s − 0.389·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.81T + 3T^{2} \) |
| 7 | \( 1 + 0.647T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 - 0.606T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 + 8.36T + 23T^{2} \) |
| 31 | \( 1 - 4.40T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 - 9.74T + 41T^{2} \) |
| 43 | \( 1 + 1.30T + 43T^{2} \) |
| 47 | \( 1 - 5.18T + 47T^{2} \) |
| 53 | \( 1 - 2.11T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 6.62T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 1.68T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 1.58T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−9.472436197747057592384180323717, −8.071709987408426186684227357811, −7.59541891018465844191264245763, −6.51622771849541359747387212312, −5.91467071263122865522569145291, −5.23617026453108237219254138048, −4.15821693906867689059270061432, −2.72712729862404241233862406756, −1.20961679332076836614305149339, 0,
1.20961679332076836614305149339, 2.72712729862404241233862406756, 4.15821693906867689059270061432, 5.23617026453108237219254138048, 5.91467071263122865522569145291, 6.51622771849541359747387212312, 7.59541891018465844191264245763, 8.071709987408426186684227357811, 9.472436197747057592384180323717
