| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s + 6·6-s + 7-s + 6·9-s − 11-s − 6·12-s − 7·13-s − 2·14-s − 4·16-s − 3·17-s − 12·18-s − 8·19-s − 3·21-s + 2·22-s − 23-s + 14·26-s − 9·27-s + 2·28-s − 5·29-s − 2·31-s + 8·32-s + 3·33-s + 6·34-s + 12·36-s + 4·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s + 2.44·6-s + 0.377·7-s + 2·9-s − 0.301·11-s − 1.73·12-s − 1.94·13-s − 0.534·14-s − 16-s − 0.727·17-s − 2.82·18-s − 1.83·19-s − 0.654·21-s + 0.426·22-s − 0.208·23-s + 2.74·26-s − 1.73·27-s + 0.377·28-s − 0.928·29-s − 0.359·31-s + 1.41·32-s + 0.522·33-s + 1.02·34-s + 2·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.73303081138440801117624820938, −6.91611918727625257998844270612, −6.58534400370676603036663207089, −5.50228320963604024452011403623, −4.79509968743593084031103946449, −4.27039865695639390758981198822, −2.35192436240687615135038228537, −1.60593211497441081297364910037, 0, 0,
1.60593211497441081297364910037, 2.35192436240687615135038228537, 4.27039865695639390758981198822, 4.79509968743593084031103946449, 5.50228320963604024452011403623, 6.58534400370676603036663207089, 6.91611918727625257998844270612, 7.73303081138440801117624820938
