L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 4·11-s − 6·13-s + 14-s + 16-s − 4·17-s + 6·19-s − 4·22-s − 6·26-s + 28-s − 6·29-s − 4·31-s + 32-s − 4·34-s − 8·37-s + 6·38-s + 10·41-s + 2·43-s − 4·44-s − 10·47-s + 49-s − 6·52-s − 14·53-s + 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 1.37·19-s − 0.852·22-s − 1.17·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.685·34-s − 1.31·37-s + 0.973·38-s + 1.56·41-s + 0.304·43-s − 0.603·44-s − 1.45·47-s + 1/7·49-s − 0.832·52-s − 1.92·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.918553946201378631298617901417, −7.59789581969321358708668328423, −6.87567123395171921294502764781, −5.81116571948616358164573245890, −5.03697469269020519729354735860, −4.72635451730829282024229729593, −3.48666647529696643095213971151, −2.63640710147708605259468073696, −1.82983565802219349369236978319, 0,
1.82983565802219349369236978319, 2.63640710147708605259468073696, 3.48666647529696643095213971151, 4.72635451730829282024229729593, 5.03697469269020519729354735860, 5.81116571948616358164573245890, 6.87567123395171921294502764781, 7.59789581969321358708668328423, 7.918553946201378631298617901417