| L(s) = 1 | + 5-s − 2·7-s − 2·13-s − 2·19-s + 25-s + 8·31-s − 2·35-s + 2·37-s − 2·43-s − 3·49-s − 6·53-s + 12·59-s − 2·61-s − 2·65-s − 4·67-s − 2·73-s + 10·79-s − 12·83-s + 6·89-s + 4·91-s − 2·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.554·13-s − 0.458·19-s + 1/5·25-s + 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.304·43-s − 3/7·49-s − 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s − 0.234·73-s + 1.12·79-s − 1.31·83-s + 0.635·89-s + 0.419·91-s − 0.205·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.84527238612872, −15.31058807020732, −14.66515184361647, −14.27303558210137, −13.51206307558259, −13.18520238580969, −12.60438941263597, −12.07245193373579, −11.49711994675907, −10.81723193535146, −10.11580682098118, −9.870265197619914, −9.235313990841747, −8.618742346115355, −7.992474624899778, −7.330122705893078, −6.546245288903881, −6.325330513022705, −5.511751432466468, −4.844701860852199, −4.216867700592509, −3.361792214262151, −2.725191039930622, −2.059299741342764, −1.043403669405913, 0,
1.043403669405913, 2.059299741342764, 2.725191039930622, 3.361792214262151, 4.216867700592509, 4.844701860852199, 5.511751432466468, 6.325330513022705, 6.546245288903881, 7.330122705893078, 7.992474624899778, 8.618742346115355, 9.235313990841747, 9.870265197619914, 10.11580682098118, 10.81723193535146, 11.49711994675907, 12.07245193373579, 12.60438941263597, 13.18520238580969, 13.51206307558259, 14.27303558210137, 14.66515184361647, 15.31058807020732, 15.84527238612872
