| L(s) = 1 | + 1.90·2-s − 2.54·3-s + 1.61·4-s − 5-s − 4.84·6-s − 0.731·8-s + 3.48·9-s − 1.90·10-s + 0.728·11-s − 4.11·12-s − 3.13·13-s + 2.54·15-s − 4.62·16-s + 2.84·17-s + 6.63·18-s + 5.77·19-s − 1.61·20-s + 1.38·22-s − 4.48·23-s + 1.86·24-s + 25-s − 5.95·26-s − 1.24·27-s + 29-s + 4.84·30-s + 5.44·31-s − 7.32·32-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 1.47·3-s + 0.807·4-s − 0.447·5-s − 1.97·6-s − 0.258·8-s + 1.16·9-s − 0.601·10-s + 0.219·11-s − 1.18·12-s − 0.869·13-s + 0.657·15-s − 1.15·16-s + 0.689·17-s + 1.56·18-s + 1.32·19-s − 0.361·20-s + 0.295·22-s − 0.934·23-s + 0.380·24-s + 0.200·25-s − 1.16·26-s − 0.238·27-s + 0.185·29-s + 0.884·30-s + 0.977·31-s − 1.29·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 3 | \( 1 + 2.54T + 3T^{2} \) |
| 11 | \( 1 - 0.728T + 11T^{2} \) |
| 13 | \( 1 + 3.13T + 13T^{2} \) |
| 17 | \( 1 - 2.84T + 17T^{2} \) |
| 19 | \( 1 - 5.77T + 19T^{2} \) |
| 23 | \( 1 + 4.48T + 23T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 + 3.86T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 - 1.56T + 47T^{2} \) |
| 53 | \( 1 - 9.73T + 53T^{2} \) |
| 59 | \( 1 + 6.37T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.17T + 67T^{2} \) |
| 71 | \( 1 + 0.416T + 71T^{2} \) |
| 73 | \( 1 - 6.24T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 3.86T + 83T^{2} \) |
| 89 | \( 1 - 3.83T + 89T^{2} \) |
| 97 | \( 1 - 3.75T + 97T^{2} \) |
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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
−7.28864061749190640390580823128, −6.60160309655215760065506123715, −5.86370968203135001924103599171, −5.50266975182223610982822078479, −4.73223852838211440800313642192, −4.31454873903927439390227186985, −3.38877672442148903011889942247, −2.59470648907501329163702963227, −1.14176201453417181116973853537, 0,
1.14176201453417181116973853537, 2.59470648907501329163702963227, 3.38877672442148903011889942247, 4.31454873903927439390227186985, 4.73223852838211440800313642192, 5.50266975182223610982822078479, 5.86370968203135001924103599171, 6.60160309655215760065506123715, 7.28864061749190640390580823128
