| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 11-s − 7·13-s + 16-s − 2·17-s − 2·20-s − 22-s + 8·23-s − 25-s + 7·26-s + 5·29-s + 4·31-s − 32-s + 2·34-s + 4·37-s + 2·40-s − 4·41-s − 8·43-s + 44-s − 8·46-s − 2·47-s + 50-s − 7·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 0.301·11-s − 1.94·13-s + 1/4·16-s − 0.485·17-s − 0.447·20-s − 0.213·22-s + 1.66·23-s − 1/5·25-s + 1.37·26-s + 0.928·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s + 0.657·37-s + 0.316·40-s − 0.624·41-s − 1.21·43-s + 0.150·44-s − 1.17·46-s − 0.291·47-s + 0.141·50-s − 0.970·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.29794521630071628867408022164, −6.95823995310811779163075905347, −6.24989160323343689314930095014, −5.01174375063564511699158702856, −4.75830932765958256241421521304, −3.71749840862877040904158561155, −2.87704159433825933818847896690, −2.20699528725113351630770660062, −0.963651736774485176630861977775, 0,
0.963651736774485176630861977775, 2.20699528725113351630770660062, 2.87704159433825933818847896690, 3.71749840862877040904158561155, 4.75830932765958256241421521304, 5.01174375063564511699158702856, 6.24989160323343689314930095014, 6.95823995310811779163075905347, 7.29794521630071628867408022164
