| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 6·11-s − 7·13-s − 4·14-s + 16-s + 7·17-s − 19-s − 6·22-s − 4·23-s − 5·25-s − 7·26-s − 4·28-s + 29-s + 32-s + 7·34-s − 4·37-s − 38-s − 3·41-s + 8·43-s − 6·44-s − 4·46-s − 12·47-s + 9·49-s − 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 1.80·11-s − 1.94·13-s − 1.06·14-s + 1/4·16-s + 1.69·17-s − 0.229·19-s − 1.27·22-s − 0.834·23-s − 25-s − 1.37·26-s − 0.755·28-s + 0.185·29-s + 0.176·32-s + 1.20·34-s − 0.657·37-s − 0.162·38-s − 0.468·41-s + 1.21·43-s − 0.904·44-s − 0.589·46-s − 1.75·47-s + 9/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 113 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−15.36322808366578, −14.81307223818082, −14.23555530838343, −13.78506117296739, −13.18630317188705, −12.64883695149418, −12.38520796613445, −12.03226299858814, −11.23245257750572, −10.32357040554738, −10.16201502941431, −9.772608898455502, −9.201303493693441, −8.043939555418914, −7.759166720020310, −7.322720921168661, −6.616734035294924, −5.841700729054973, −5.626080731114012, −4.861019083846042, −4.376662807384980, −3.271391371141297, −3.098928431313337, −2.478268768488238, −1.660906896424266, 0, 0,
1.660906896424266, 2.478268768488238, 3.098928431313337, 3.271391371141297, 4.376662807384980, 4.861019083846042, 5.626080731114012, 5.841700729054973, 6.616734035294924, 7.322720921168661, 7.759166720020310, 8.043939555418914, 9.201303493693441, 9.772608898455502, 10.16201502941431, 10.32357040554738, 11.23245257750572, 12.03226299858814, 12.38520796613445, 12.64883695149418, 13.18630317188705, 13.78506117296739, 14.23555530838343, 14.81307223818082, 15.36322808366578
