| L(s) = 1 | − 3·2-s + 4-s + 5·5-s − 22·7-s + 21·8-s − 15·10-s − 60·11-s − 31·13-s + 66·14-s − 71·16-s − 17·17-s − 61·19-s + 5·20-s + 180·22-s + 78·23-s + 25·25-s + 93·26-s − 22·28-s − 69·29-s − 31·31-s + 45·32-s + 51·34-s − 110·35-s + 56·37-s + 183·38-s + 105·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s + 0.447·5-s − 1.18·7-s + 0.928·8-s − 0.474·10-s − 1.64·11-s − 0.661·13-s + 1.25·14-s − 1.10·16-s − 0.242·17-s − 0.736·19-s + 0.0559·20-s + 1.74·22-s + 0.707·23-s + 1/5·25-s + 0.701·26-s − 0.148·28-s − 0.441·29-s − 0.179·31-s + 0.248·32-s + 0.257·34-s − 0.531·35-s + 0.248·37-s + 0.781·38-s + 0.415·40-s + 0.0228·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3646647491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3646647491\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 17 | \( 1 + p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 31 T + p^{3} T^{2} \) |
| 19 | \( 1 + 61 T + p^{3} T^{2} \) |
| 23 | \( 1 - 78 T + p^{3} T^{2} \) |
| 29 | \( 1 + 69 T + p^{3} T^{2} \) |
| 31 | \( 1 + p T + p^{3} T^{2} \) |
| 37 | \( 1 - 56 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 538 T + p^{3} T^{2} \) |
| 47 | \( 1 - 465 T + p^{3} T^{2} \) |
| 53 | \( 1 + 723 T + p^{3} T^{2} \) |
| 59 | \( 1 - 753 T + p^{3} T^{2} \) |
| 61 | \( 1 - 35 T + p^{3} T^{2} \) |
| 67 | \( 1 + 322 T + p^{3} T^{2} \) |
| 71 | \( 1 - 99 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1123 T + p^{3} T^{2} \) |
| 79 | \( 1 - 488 T + p^{3} T^{2} \) |
| 83 | \( 1 - 852 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1215 T + p^{3} T^{2} \) |
| 97 | \( 1 + 601 T + p^{3} T^{2} \) |
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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
−9.944766770502961923522907547030, −9.184748306235144409894056342288, −8.373882653281523124203635228799, −7.47417548472793914915300253860, −6.69655275525163087708879067751, −5.53165823352917759061316795678, −4.59590987352733222571296164938, −3.08458140361196906144099413704, −2.05410476319478175483653685954, −0.38009715439859127411271061923,
0.38009715439859127411271061923, 2.05410476319478175483653685954, 3.08458140361196906144099413704, 4.59590987352733222571296164938, 5.53165823352917759061316795678, 6.69655275525163087708879067751, 7.47417548472793914915300253860, 8.373882653281523124203635228799, 9.184748306235144409894056342288, 9.944766770502961923522907547030
