| L(s) = 1 | + 1.26·3-s − 1.77·5-s − 1.40·9-s + 0.318·11-s + 0.205·13-s − 2.24·15-s − 17-s + 5.45·19-s + 9.43·23-s − 1.85·25-s − 5.56·27-s − 8.61·29-s + 4.94·31-s + 0.402·33-s + 7.64·37-s + 0.260·39-s + 1.44·41-s − 1.98·43-s + 2.48·45-s − 1.56·47-s − 1.26·51-s − 13.5·53-s − 0.564·55-s + 6.90·57-s + 13.2·59-s + 3.37·61-s − 0.365·65-s + ⋯ |
| L(s) = 1 | + 0.729·3-s − 0.793·5-s − 0.467·9-s + 0.0959·11-s + 0.0571·13-s − 0.579·15-s − 0.242·17-s + 1.25·19-s + 1.96·23-s − 0.370·25-s − 1.07·27-s − 1.60·29-s + 0.888·31-s + 0.0700·33-s + 1.25·37-s + 0.0416·39-s + 0.225·41-s − 0.302·43-s + 0.370·45-s − 0.228·47-s − 0.176·51-s − 1.85·53-s − 0.0761·55-s + 0.913·57-s + 1.71·59-s + 0.431·61-s − 0.0453·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.923527292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923527292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.26T + 3T^{2} \) |
| 5 | \( 1 + 1.77T + 5T^{2} \) |
| 11 | \( 1 - 0.318T + 11T^{2} \) |
| 13 | \( 1 - 0.205T + 13T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 - 9.43T + 23T^{2} \) |
| 29 | \( 1 + 8.61T + 29T^{2} \) |
| 31 | \( 1 - 4.94T + 31T^{2} \) |
| 37 | \( 1 - 7.64T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 + 1.98T + 43T^{2} \) |
| 47 | \( 1 + 1.56T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 - 3.86T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 0.00620T + 89T^{2} \) |
| 97 | \( 1 - 4.93T + 97T^{2} \) |
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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
−8.562810630973310878637636408320, −7.86989897993242159194584187615, −7.40332831515169569416557151092, −6.50311814105352575721622381441, −5.50440371670438408391081985405, −4.75277361801580370315635727664, −3.66815107272191255448819501807, −3.19908917824545086579352607243, −2.21431248410130303331426350607, −0.794758986820798593238873719151,
0.794758986820798593238873719151, 2.21431248410130303331426350607, 3.19908917824545086579352607243, 3.66815107272191255448819501807, 4.75277361801580370315635727664, 5.50440371670438408391081985405, 6.50311814105352575721622381441, 7.40332831515169569416557151092, 7.86989897993242159194584187615, 8.562810630973310878637636408320
