| L(s) = 1 | + 1.34·3-s + 0.901·5-s + 0.205·7-s − 1.18·9-s − 0.815·11-s − 0.909·13-s + 1.21·15-s − 7.34·17-s + 19-s + 0.277·21-s − 0.663·23-s − 4.18·25-s − 5.63·27-s + 6.35·29-s + 0.965·31-s − 1.09·33-s + 0.185·35-s + 8.11·37-s − 1.22·39-s − 11.6·41-s − 3.30·43-s − 1.06·45-s + 6.76·47-s − 6.95·49-s − 9.90·51-s − 53-s − 0.735·55-s + ⋯ |
| L(s) = 1 | + 0.778·3-s + 0.403·5-s + 0.0778·7-s − 0.393·9-s − 0.245·11-s − 0.252·13-s + 0.313·15-s − 1.78·17-s + 0.229·19-s + 0.0606·21-s − 0.138·23-s − 0.837·25-s − 1.08·27-s + 1.18·29-s + 0.173·31-s − 0.191·33-s + 0.0313·35-s + 1.33·37-s − 0.196·39-s − 1.82·41-s − 0.503·43-s − 0.158·45-s + 0.987·47-s − 0.993·49-s − 1.38·51-s − 0.137·53-s − 0.0991·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 - 0.901T + 5T^{2} \) |
| 7 | \( 1 - 0.205T + 7T^{2} \) |
| 11 | \( 1 + 0.815T + 11T^{2} \) |
| 13 | \( 1 + 0.909T + 13T^{2} \) |
| 17 | \( 1 + 7.34T + 17T^{2} \) |
| 23 | \( 1 + 0.663T + 23T^{2} \) |
| 29 | \( 1 - 6.35T + 29T^{2} \) |
| 31 | \( 1 - 0.965T + 31T^{2} \) |
| 37 | \( 1 - 8.11T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 3.30T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 5.28T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 4.10T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 3.72T + 89T^{2} \) |
| 97 | \( 1 - 3.36T + 97T^{2} \) |
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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
−8.146189563488844155189585532071, −7.52306284115937252346818017584, −6.53158982631704527246690504371, −6.00090811355652233693022043573, −4.95151120291305082401368480550, −4.29775430494132602459796021736, −3.19620036168010963855189194836, −2.51285760564607479405524894208, −1.71542199974952972624745563777, 0,
1.71542199974952972624745563777, 2.51285760564607479405524894208, 3.19620036168010963855189194836, 4.29775430494132602459796021736, 4.95151120291305082401368480550, 6.00090811355652233693022043573, 6.53158982631704527246690504371, 7.52306284115937252346818017584, 8.146189563488844155189585532071
