| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 11-s + 4·13-s + 16-s + 2·17-s + 6·19-s + 2·20-s + 22-s + 2·23-s − 25-s + 4·26-s + 2·29-s + 8·31-s + 32-s + 2·34-s − 2·37-s + 6·38-s + 2·40-s − 2·41-s − 2·43-s + 44-s + 2·46-s + 2·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 0.301·11-s + 1.10·13-s + 1/4·16-s + 0.485·17-s + 1.37·19-s + 0.447·20-s + 0.213·22-s + 0.417·23-s − 1/5·25-s + 0.784·26-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s − 0.328·37-s + 0.973·38-s + 0.316·40-s − 0.312·41-s − 0.304·43-s + 0.150·44-s + 0.294·46-s + 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.062076218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.062076218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.55681813847005856187427135991, −6.79923493510247629545251393096, −6.15158471182841855400365242253, −5.69276000012681049628626494586, −5.00735722361764893107385149224, −4.23232022692161449850766373388, −3.33365417041840785728112437626, −2.83066461663364441278499665394, −1.69008397864184290335637215528, −1.06738870446855368402014094401,
1.06738870446855368402014094401, 1.69008397864184290335637215528, 2.83066461663364441278499665394, 3.33365417041840785728112437626, 4.23232022692161449850766373388, 5.00735722361764893107385149224, 5.69276000012681049628626494586, 6.15158471182841855400365242253, 6.79923493510247629545251393096, 7.55681813847005856187427135991
