| L(s) = 1 | + 5-s + 2·7-s − 11-s + 4·17-s − 6·19-s + 25-s + 4·29-s + 4·31-s + 2·35-s − 10·37-s + 10·43-s − 3·49-s + 2·53-s − 55-s + 4·59-s + 10·61-s − 8·67-s − 8·71-s + 14·73-s − 2·77-s − 10·79-s + 4·83-s + 4·85-s − 10·89-s − 6·95-s + 10·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.301·11-s + 0.970·17-s − 1.37·19-s + 1/5·25-s + 0.742·29-s + 0.718·31-s + 0.338·35-s − 1.64·37-s + 1.52·43-s − 3/7·49-s + 0.274·53-s − 0.134·55-s + 0.520·59-s + 1.28·61-s − 0.977·67-s − 0.949·71-s + 1.63·73-s − 0.227·77-s − 1.12·79-s + 0.439·83-s + 0.433·85-s − 1.05·89-s − 0.615·95-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.335678244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.335678244\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.60426744366798, −12.16450482434150, −11.70922821907639, −11.19228622815064, −10.70588051485363, −10.29587168477199, −10.02962057797297, −9.423511356960377, −8.732922044208405, −8.553186285212816, −8.045871885008110, −7.554525692438696, −7.011692378539163, −6.537287677835192, −6.003070911857499, −5.502171358658906, −5.117185184649459, −4.482466764730936, −4.165589528491474, −3.394459176019814, −2.884280713167789, −2.220623682908858, −1.827918624814188, −1.126157939433746, −0.5022038341590072,
0.5022038341590072, 1.126157939433746, 1.827918624814188, 2.220623682908858, 2.884280713167789, 3.394459176019814, 4.165589528491474, 4.482466764730936, 5.117185184649459, 5.502171358658906, 6.003070911857499, 6.537287677835192, 7.011692378539163, 7.554525692438696, 8.045871885008110, 8.553186285212816, 8.732922044208405, 9.423511356960377, 10.02962057797297, 10.29587168477199, 10.70588051485363, 11.19228622815064, 11.70922821907639, 12.16450482434150, 12.60426744366798
