| L(s) = 1 | + 1.12·2-s + 2.23·3-s − 0.730·4-s + 0.792·5-s + 2.52·6-s − 3.80·7-s − 3.07·8-s + 2.01·9-s + 0.893·10-s − 11-s − 1.63·12-s − 4.28·14-s + 1.77·15-s − 2.00·16-s − 3.83·17-s + 2.26·18-s − 7.92·19-s − 0.578·20-s − 8.52·21-s − 1.12·22-s − 2.44·23-s − 6.88·24-s − 4.37·25-s − 2.20·27-s + 2.78·28-s − 2.56·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.796·2-s + 1.29·3-s − 0.365·4-s + 0.354·5-s + 1.03·6-s − 1.43·7-s − 1.08·8-s + 0.671·9-s + 0.282·10-s − 0.301·11-s − 0.472·12-s − 1.14·14-s + 0.458·15-s − 0.501·16-s − 0.931·17-s + 0.534·18-s − 1.81·19-s − 0.129·20-s − 1.86·21-s − 0.240·22-s − 0.510·23-s − 1.40·24-s − 0.874·25-s − 0.424·27-s + 0.525·28-s − 0.476·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 0.792T + 5T^{2} \) |
| 7 | \( 1 + 3.80T + 7T^{2} \) |
| 17 | \( 1 + 3.83T + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 - 2.32T + 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 - 7.74T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 - 4.49T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 + 6.82T + 61T^{2} \) |
| 67 | \( 1 - 4.97T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 3.48T + 79T^{2} \) |
| 83 | \( 1 + 6.28T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−8.931134283702196525733350428688, −8.281610894200505039510256765109, −7.24138310862616076660547256428, −6.16508345392726620115278721396, −5.82767955984573138976068481597, −4.21356301544326811920910800395, −4.00149773499433914560858530034, −2.78182689584639267195361696144, −2.36281618707039146518093179389, 0,
2.36281618707039146518093179389, 2.78182689584639267195361696144, 4.00149773499433914560858530034, 4.21356301544326811920910800395, 5.82767955984573138976068481597, 6.16508345392726620115278721396, 7.24138310862616076660547256428, 8.281610894200505039510256765109, 8.931134283702196525733350428688
