| L(s) = 1 | + 2.70·2-s − 2.50·3-s + 5.29·4-s + 0.533·5-s − 6.77·6-s + 0.354·7-s + 8.89·8-s + 3.29·9-s + 1.44·10-s + 4.18·11-s − 13.2·12-s − 3.72·13-s + 0.958·14-s − 1.33·15-s + 13.4·16-s − 6.46·17-s + 8.88·18-s − 1.31·19-s + 2.82·20-s − 0.890·21-s + 11.3·22-s − 4.10·23-s − 22.3·24-s − 4.71·25-s − 10.0·26-s − 0.728·27-s + 1.87·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 1.44·3-s + 2.64·4-s + 0.238·5-s − 2.76·6-s + 0.134·7-s + 3.14·8-s + 1.09·9-s + 0.455·10-s + 1.26·11-s − 3.83·12-s − 1.03·13-s + 0.256·14-s − 0.345·15-s + 3.35·16-s − 1.56·17-s + 2.09·18-s − 0.300·19-s + 0.631·20-s − 0.194·21-s + 2.41·22-s − 0.856·23-s − 4.55·24-s − 0.943·25-s − 1.97·26-s − 0.140·27-s + 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.501908596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.501908596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 - 0.533T + 5T^{2} \) |
| 7 | \( 1 - 0.354T + 7T^{2} \) |
| 11 | \( 1 - 4.18T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 + 1.31T + 19T^{2} \) |
| 23 | \( 1 + 4.10T + 23T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 0.673T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 7.76T + 73T^{2} \) |
| 79 | \( 1 + 1.17T + 79T^{2} \) |
| 83 | \( 1 - 6.25T + 83T^{2} \) |
| 89 | \( 1 - 3.80T + 89T^{2} \) |
| 97 | \( 1 + 9.91T + 97T^{2} \) |
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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
−12.19218581549686301341673671304, −11.41244554203507788752074829581, −10.98744154474747761661737303336, −9.629330397344675315824190414139, −7.45832247345196531925396565274, −6.38500235591686135993432646918, −5.97492081923480481647941477588, −4.76770483848132832529271571907, −4.08864130435105247531414129255, −2.12048939619168316958162889512,
2.12048939619168316958162889512, 4.08864130435105247531414129255, 4.76770483848132832529271571907, 5.97492081923480481647941477588, 6.38500235591686135993432646918, 7.45832247345196531925396565274, 9.629330397344675315824190414139, 10.98744154474747761661737303336, 11.41244554203507788752074829581, 12.19218581549686301341673671304
