| L(s) = 1 | + 3-s + 4·5-s − 4·7-s + 9-s − 2·13-s + 4·15-s + 17-s − 2·19-s − 4·21-s − 4·23-s + 11·25-s + 27-s − 6·29-s + 8·31-s − 16·35-s − 2·37-s − 2·39-s + 6·41-s + 10·43-s + 4·45-s − 2·47-s + 9·49-s + 51-s + 10·53-s − 2·57-s − 10·59-s + 8·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s + 1.03·15-s + 0.242·17-s − 0.458·19-s − 0.872·21-s − 0.834·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 2.70·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s + 1.52·43-s + 0.596·45-s − 0.291·47-s + 9/7·49-s + 0.140·51-s + 1.37·53-s − 0.264·57-s − 1.30·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.488813658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.488813658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.76749313629114, −13.29880505468325, −12.92153272501838, −12.49265232593482, −12.08216627741591, −11.21313129419335, −10.48399351424477, −10.17821660419535, −9.769122520622683, −9.405169101429864, −9.011099277536046, −8.462571122339688, −7.654253710804259, −7.184353720684315, −6.486672348480956, −6.224784652184003, −5.714717653606344, −5.201391026316674, −4.366253011863411, −3.803547515030035, −3.076494327066852, −2.464675808775697, −2.271182831798017, −1.390990726740157, −0.5492736011660251,
0.5492736011660251, 1.390990726740157, 2.271182831798017, 2.464675808775697, 3.076494327066852, 3.803547515030035, 4.366253011863411, 5.201391026316674, 5.714717653606344, 6.224784652184003, 6.486672348480956, 7.184353720684315, 7.654253710804259, 8.462571122339688, 9.011099277536046, 9.405169101429864, 9.769122520622683, 10.17821660419535, 10.48399351424477, 11.21313129419335, 12.08216627741591, 12.49265232593482, 12.92153272501838, 13.29880505468325, 13.76749313629114
