| L(s) = 1 | + 2.73·2-s + 1.15·3-s + 5.47·4-s + 1.87·5-s + 3.16·6-s + 9.50·8-s − 1.66·9-s + 5.13·10-s − 2.29·11-s + 6.32·12-s + 2.17·15-s + 15.0·16-s + 6.07·17-s − 4.54·18-s − 5.15·19-s + 10.2·20-s − 6.28·22-s + 4.41·23-s + 10.9·24-s − 1.47·25-s − 5.39·27-s + 7.50·29-s + 5.93·30-s + 4.33·31-s + 22.0·32-s − 2.65·33-s + 16.6·34-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 0.667·3-s + 2.73·4-s + 0.839·5-s + 1.29·6-s + 3.36·8-s − 0.554·9-s + 1.62·10-s − 0.693·11-s + 1.82·12-s + 0.560·15-s + 3.75·16-s + 1.47·17-s − 1.07·18-s − 1.18·19-s + 2.29·20-s − 1.34·22-s + 0.921·23-s + 2.24·24-s − 0.295·25-s − 1.03·27-s + 1.39·29-s + 1.08·30-s + 0.778·31-s + 3.90·32-s − 0.462·33-s + 2.84·34-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
| \(L(1)\) |
\(\approx\) |
\(11.47844177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.47844177\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.73T + 2T^{2} \) |
| 3 | \( 1 - 1.15T + 3T^{2} \) |
| 5 | \( 1 - 1.87T + 5T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 17 | \( 1 - 6.07T + 17T^{2} \) |
| 19 | \( 1 + 5.15T + 19T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 - 7.50T + 29T^{2} \) |
| 31 | \( 1 - 4.33T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 - 2.45T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 + 8.90T + 47T^{2} \) |
| 53 | \( 1 - 8.10T + 53T^{2} \) |
| 59 | \( 1 + 7.13T + 59T^{2} \) |
| 61 | \( 1 + 8.00T + 61T^{2} \) |
| 67 | \( 1 - 1.15T + 67T^{2} \) |
| 71 | \( 1 - 5.11T + 71T^{2} \) |
| 73 | \( 1 + 1.96T + 73T^{2} \) |
| 79 | \( 1 - 3.81T + 79T^{2} \) |
| 83 | \( 1 - 2.49T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 2.51T + 97T^{2} \) |
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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
−7.78535484635518835883090355123, −6.75830595730628329730983243203, −6.19153511928703768243820481403, −5.62100351288396128796777014511, −5.04687424303513330573436947392, −4.31509346675617922076647481817, −3.41274816386459325172920830249, −2.77063410054549385535896700509, −2.37526080034769565449732499993, −1.33946489288962216060031811116,
1.33946489288962216060031811116, 2.37526080034769565449732499993, 2.77063410054549385535896700509, 3.41274816386459325172920830249, 4.31509346675617922076647481817, 5.04687424303513330573436947392, 5.62100351288396128796777014511, 6.19153511928703768243820481403, 6.75830595730628329730983243203, 7.78535484635518835883090355123
