| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 7-s + 9-s + 4·12-s + 2·13-s + 2·14-s − 4·16-s − 5·17-s + 2·18-s + 8·19-s + 2·21-s − 23-s + 4·26-s − 4·27-s + 2·28-s − 5·29-s − 5·31-s − 8·32-s − 10·34-s + 2·36-s + 7·37-s + 16·38-s + 4·39-s − 7·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 0.377·7-s + 1/3·9-s + 1.15·12-s + 0.554·13-s + 0.534·14-s − 16-s − 1.21·17-s + 0.471·18-s + 1.83·19-s + 0.436·21-s − 0.208·23-s + 0.784·26-s − 0.769·27-s + 0.377·28-s − 0.928·29-s − 0.898·31-s − 1.41·32-s − 1.71·34-s + 1/3·36-s + 1.15·37-s + 2.59·38-s + 0.640·39-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.064745297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.064745297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−11.21645730685170637539946705503, −9.635573568718492337153683237964, −8.968247392855360442850494902852, −8.021393070095707242610426057673, −7.06279245617742916674477673238, −5.90881828705332726272674122854, −4.99590413546970297987417470463, −3.89354701982092128675905563532, −3.17036354408394678886960890665, −2.05368682964998244414756512759,
2.05368682964998244414756512759, 3.17036354408394678886960890665, 3.89354701982092128675905563532, 4.99590413546970297987417470463, 5.90881828705332726272674122854, 7.06279245617742916674477673238, 8.021393070095707242610426057673, 8.968247392855360442850494902852, 9.635573568718492337153683237964, 11.21645730685170637539946705503
