| L(s) = 1 | − 2.41·2-s − 1.41·3-s + 3.82·4-s + 3.82·5-s + 3.41·6-s − 4.41·8-s − 0.999·9-s − 9.24·10-s + 3.41·11-s − 5.41·12-s − 5.41·15-s + 2.99·16-s − 0.171·17-s + 2.41·18-s + 6·19-s + 14.6·20-s − 8.24·22-s + 1.41·23-s + 6.24·24-s + 9.65·25-s + 5.65·27-s + 9.82·29-s + 13.0·30-s + 5.41·31-s + 1.58·32-s − 4.82·33-s + 0.414·34-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.816·3-s + 1.91·4-s + 1.71·5-s + 1.39·6-s − 1.56·8-s − 0.333·9-s − 2.92·10-s + 1.02·11-s − 1.56·12-s − 1.39·15-s + 0.749·16-s − 0.0416·17-s + 0.569·18-s + 1.37·19-s + 3.27·20-s − 1.75·22-s + 0.294·23-s + 1.27·24-s + 1.93·25-s + 1.08·27-s + 1.82·29-s + 2.38·30-s + 0.972·31-s + 0.280·32-s − 0.840·33-s + 0.0710·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.240053210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.240053210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 17 | \( 1 + 0.171T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 9.82T + 29T^{2} \) |
| 31 | \( 1 - 5.41T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 5.82T + 41T^{2} \) |
| 43 | \( 1 - 0.585T + 43T^{2} \) |
| 47 | \( 1 - 7.65T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 0.343T + 71T^{2} \) |
| 73 | \( 1 - 0.656T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 7.31T + 89T^{2} \) |
| 97 | \( 1 + 5.17T + 97T^{2} \) |
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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
−7.943096179944719793363925231302, −7.03359509175422905629862886051, −6.45174112981195242314564858631, −6.07247154525041842185906693227, −5.36451236302739369117253248483, −4.50499646001724715689051813237, −2.92150443729968954435781723394, −2.39398753943418018121915483720, −1.16787507405949207328540395731, −0.948817116965434503947995427287,
0.948817116965434503947995427287, 1.16787507405949207328540395731, 2.39398753943418018121915483720, 2.92150443729968954435781723394, 4.50499646001724715689051813237, 5.36451236302739369117253248483, 6.07247154525041842185906693227, 6.45174112981195242314564858631, 7.03359509175422905629862886051, 7.943096179944719793363925231302
