| L(s) = 1 | + 1.56·2-s − 3-s + 0.455·4-s − 1.56·6-s − 0.875·7-s − 2.42·8-s + 9-s − 2.76·11-s − 0.455·12-s + 4.96·13-s − 1.37·14-s − 4.70·16-s − 0.786·17-s + 1.56·18-s + 2.72·19-s + 0.875·21-s − 4.33·22-s + 5.11·23-s + 2.42·24-s + 7.77·26-s − 27-s − 0.398·28-s − 8.80·29-s − 8.03·31-s − 2.52·32-s + 2.76·33-s − 1.23·34-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.577·3-s + 0.227·4-s − 0.639·6-s − 0.331·7-s − 0.855·8-s + 0.333·9-s − 0.833·11-s − 0.131·12-s + 1.37·13-s − 0.366·14-s − 1.17·16-s − 0.190·17-s + 0.369·18-s + 0.625·19-s + 0.191·21-s − 0.923·22-s + 1.06·23-s + 0.494·24-s + 1.52·26-s − 0.192·27-s − 0.0753·28-s − 1.63·29-s − 1.44·31-s − 0.446·32-s + 0.481·33-s − 0.211·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.036665951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.036665951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 - T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 7 | \( 1 + 0.875T + 7T^{2} \) |
| 11 | \( 1 + 2.76T + 11T^{2} \) |
| 13 | \( 1 - 4.96T + 13T^{2} \) |
| 17 | \( 1 + 0.786T + 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 + 8.80T + 29T^{2} \) |
| 31 | \( 1 + 8.03T + 31T^{2} \) |
| 37 | \( 1 - 3.04T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 43 | \( 1 - 2.67T + 43T^{2} \) |
| 47 | \( 1 - 0.693T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 - 3.58T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 5.44T + 73T^{2} \) |
| 79 | \( 1 + 8.34T + 79T^{2} \) |
| 83 | \( 1 + 0.591T + 83T^{2} \) |
| 89 | \( 1 + 9.94T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.67547252447124914132340134677, −6.86378637193262191652769659915, −6.24996171351241121498526003199, −5.44604697205432906951474763954, −5.30654184492032880237269429684, −4.29103751328791901704720517789, −3.60761337964187914040554143251, −3.05008137151299728388262905702, −1.90547283647648496364585658740, −0.60407783172681225277799708625,
0.60407783172681225277799708625, 1.90547283647648496364585658740, 3.05008137151299728388262905702, 3.60761337964187914040554143251, 4.29103751328791901704720517789, 5.30654184492032880237269429684, 5.44604697205432906951474763954, 6.24996171351241121498526003199, 6.86378637193262191652769659915, 7.67547252447124914132340134677
