| L(s) = 1 | + 0.669·2-s − 1.55·4-s + 2.76·5-s − 1.51·7-s − 2.37·8-s + 1.85·10-s − 1.48·13-s − 1.01·14-s + 1.51·16-s + 4.72·17-s + 3.55·19-s − 4.28·20-s − 2.09·23-s + 2.64·25-s − 0.994·26-s + 2.34·28-s + 7.85·29-s − 8.76·31-s + 5.76·32-s + 3.16·34-s − 4.17·35-s − 5.15·37-s + 2.37·38-s − 6.57·40-s + 7.02·41-s − 2.91·43-s − 1.40·46-s + ⋯ |
| L(s) = 1 | + 0.473·2-s − 0.775·4-s + 1.23·5-s − 0.570·7-s − 0.840·8-s + 0.585·10-s − 0.411·13-s − 0.270·14-s + 0.377·16-s + 1.14·17-s + 0.814·19-s − 0.959·20-s − 0.436·23-s + 0.528·25-s − 0.195·26-s + 0.442·28-s + 1.45·29-s − 1.57·31-s + 1.01·32-s + 0.543·34-s − 0.705·35-s − 0.847·37-s + 0.385·38-s − 1.03·40-s + 1.09·41-s − 0.444·43-s − 0.206·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.188845667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.188845667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.669T + 2T^{2} \) |
| 5 | \( 1 - 2.76T + 5T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 4.72T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 + 2.09T + 23T^{2} \) |
| 29 | \( 1 - 7.85T + 29T^{2} \) |
| 31 | \( 1 + 8.76T + 31T^{2} \) |
| 37 | \( 1 + 5.15T + 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 + 2.91T + 43T^{2} \) |
| 47 | \( 1 + 2.17T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 6.72T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 4.49T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 3.34T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 2.22T + 97T^{2} \) |
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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
−8.810989781207519621197179249093, −7.960576447972142127316692624578, −6.99131873233252625282620250780, −6.16811827182808883712013330919, −5.44496701008033158198053063791, −5.10779803915741793891217359277, −3.88122864253773006642904330179, −3.18361063817497759198545129897, −2.17492431679844958963655502007, −0.839296407225174944784502255083,
0.839296407225174944784502255083, 2.17492431679844958963655502007, 3.18361063817497759198545129897, 3.88122864253773006642904330179, 5.10779803915741793891217359277, 5.44496701008033158198053063791, 6.16811827182808883712013330919, 6.99131873233252625282620250780, 7.960576447972142127316692624578, 8.810989781207519621197179249093
