| L(s) = 1 | − 1.36·2-s − 1.31·3-s − 0.146·4-s − 3.05·5-s + 1.78·6-s + 2.92·8-s − 1.27·9-s + 4.16·10-s + 1.55·11-s + 0.192·12-s + 4.01·15-s − 3.68·16-s + 5.82·17-s + 1.73·18-s + 1.44·19-s + 0.448·20-s − 2.11·22-s + 6.27·23-s − 3.83·24-s + 4.35·25-s + 5.61·27-s − 9.88·29-s − 5.46·30-s + 1.52·31-s − 0.827·32-s − 2.03·33-s − 7.92·34-s + ⋯ |
| L(s) = 1 | − 0.962·2-s − 0.758·3-s − 0.0732·4-s − 1.36·5-s + 0.729·6-s + 1.03·8-s − 0.425·9-s + 1.31·10-s + 0.467·11-s + 0.0555·12-s + 1.03·15-s − 0.921·16-s + 1.41·17-s + 0.409·18-s + 0.331·19-s + 0.100·20-s − 0.450·22-s + 1.30·23-s − 0.783·24-s + 0.871·25-s + 1.08·27-s − 1.83·29-s − 0.998·30-s + 0.274·31-s − 0.146·32-s − 0.354·33-s − 1.35·34-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
| \(L(1)\) |
\(\approx\) |
\(0.6070813572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6070813572\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.36T + 2T^{2} \) |
| 3 | \( 1 + 1.31T + 3T^{2} \) |
| 5 | \( 1 + 3.05T + 5T^{2} \) |
| 11 | \( 1 - 1.55T + 11T^{2} \) |
| 17 | \( 1 - 5.82T + 17T^{2} \) |
| 19 | \( 1 - 1.44T + 19T^{2} \) |
| 23 | \( 1 - 6.27T + 23T^{2} \) |
| 29 | \( 1 + 9.88T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 7.75T + 37T^{2} \) |
| 41 | \( 1 - 7.17T + 41T^{2} \) |
| 43 | \( 1 - 5.03T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 7.77T + 53T^{2} \) |
| 59 | \( 1 - 2.79T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 0.683T + 67T^{2} \) |
| 71 | \( 1 + 0.582T + 71T^{2} \) |
| 73 | \( 1 - 4.32T + 73T^{2} \) |
| 79 | \( 1 - 6.41T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 2.06T + 89T^{2} \) |
| 97 | \( 1 - 5.78T + 97T^{2} \) |
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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
−7.77799430140600323818650744699, −7.42115301024150013754080997269, −6.66056990416909264914690591818, −5.59646162678170009720642632884, −5.17239078022982344213154077533, −4.13999788988304048215883270385, −3.70806454075944667608501410854, −2.62139262555994266613587182218, −1.07890719711275483803514336333, −0.60105392715023852659424359231,
0.60105392715023852659424359231, 1.07890719711275483803514336333, 2.62139262555994266613587182218, 3.70806454075944667608501410854, 4.13999788988304048215883270385, 5.17239078022982344213154077533, 5.59646162678170009720642632884, 6.66056990416909264914690591818, 7.42115301024150013754080997269, 7.77799430140600323818650744699
