| L(s) = 1 | − 4·4-s + 84·11-s + 16·16-s − 118·19-s − 240·29-s + 94·31-s + 252·41-s − 336·44-s + 670·49-s + 888·59-s + 442·61-s − 64·64-s + 1.38e3·71-s + 472·76-s − 1.33e3·79-s + 2.17e3·89-s − 264·101-s + 3.47e3·109-s + 960·116-s + 2.63e3·121-s − 376·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2.30·11-s + 1/4·16-s − 1.42·19-s − 1.53·29-s + 0.544·31-s + 0.959·41-s − 1.15·44-s + 1.95·49-s + 1.95·59-s + 0.927·61-s − 1/8·64-s + 2.30·71-s + 0.712·76-s − 1.89·79-s + 2.58·89-s − 0.260·101-s + 3.04·109-s + 0.768·116-s + 1.97·121-s − 0.272·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.934783885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.934783885\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 42 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1177 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 59 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24253 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 47 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 32662 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 126 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 127330 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 186910 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 251327 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 444 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 221 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 312082 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 690 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 489842 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 665 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1137949 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1086 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 558590 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.210191206923642824138921849973, −9.147615805329793846652826920429, −8.630777598559242197788999732232, −8.497347995772085792498050979716, −7.77779682647120438344187375353, −7.45305904177883104142022706040, −6.81371120112275909901177677097, −6.65174236074394373347402543304, −6.20251906897520799078319469150, −5.68198584630753784983959077845, −5.36727860747056066857200244380, −4.63651095380745371319392061376, −4.17332979981682554490961887896, −3.89226455713689136770562952240, −3.65848701001642783059719922947, −2.81765928628447428332018317483, −2.05021787367763189025816626853, −1.80384376833010856932727773588, −0.78277061714038785617155077545, −0.65807779595775183403601786108,
0.65807779595775183403601786108, 0.78277061714038785617155077545, 1.80384376833010856932727773588, 2.05021787367763189025816626853, 2.81765928628447428332018317483, 3.65848701001642783059719922947, 3.89226455713689136770562952240, 4.17332979981682554490961887896, 4.63651095380745371319392061376, 5.36727860747056066857200244380, 5.68198584630753784983959077845, 6.20251906897520799078319469150, 6.65174236074394373347402543304, 6.81371120112275909901177677097, 7.45305904177883104142022706040, 7.77779682647120438344187375353, 8.497347995772085792498050979716, 8.630777598559242197788999732232, 9.147615805329793846652826920429, 9.210191206923642824138921849973
