| L(s) = 1 | − 3-s − 2·4-s + 7-s − 2·9-s − 11-s + 2·12-s − 13-s + 4·16-s − 17-s + 2·19-s − 21-s − 23-s + 5·27-s − 2·28-s + 7·29-s + 4·31-s + 33-s + 4·36-s − 8·37-s + 39-s − 6·41-s + 8·43-s + 2·44-s + 7·47-s − 4·48-s + 49-s + 51-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.377·7-s − 2/3·9-s − 0.301·11-s + 0.577·12-s − 0.277·13-s + 16-s − 0.242·17-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.962·27-s − 0.377·28-s + 1.29·29-s + 0.718·31-s + 0.174·33-s + 2/3·36-s − 1.31·37-s + 0.160·39-s − 0.937·41-s + 1.21·43-s + 0.301·44-s + 1.02·47-s − 0.577·48-s + 1/7·49-s + 0.140·51-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 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.227424872801678942008594600403, −7.44488666107612493453369797646, −6.46305776873434412629623573176, −5.70551226127103814314548639498, −5.02837085429780677206654130848, −4.53638690439984486481221985630, −3.48098161163945720129544839182, −2.55208017739021272500029411580, −1.11047280102303688568605393583, 0,
1.11047280102303688568605393583, 2.55208017739021272500029411580, 3.48098161163945720129544839182, 4.53638690439984486481221985630, 5.02837085429780677206654130848, 5.70551226127103814314548639498, 6.46305776873434412629623573176, 7.44488666107612493453369797646, 8.227424872801678942008594600403
