| L(s) = 1 | + 2·2-s + 8·3-s + 4·4-s + 16·6-s + 8·8-s + 37·9-s − 28·11-s + 32·12-s + 18·13-s + 16·16-s + 74·17-s + 74·18-s − 80·19-s − 56·22-s + 112·23-s + 64·24-s + 36·26-s + 80·27-s + 190·29-s − 72·31-s + 32·32-s − 224·33-s + 148·34-s + 148·36-s + 346·37-s − 160·38-s + 144·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 1/2·4-s + 1.08·6-s + 0.353·8-s + 1.37·9-s − 0.767·11-s + 0.769·12-s + 0.384·13-s + 1/4·16-s + 1.05·17-s + 0.968·18-s − 0.965·19-s − 0.542·22-s + 1.01·23-s + 0.544·24-s + 0.271·26-s + 0.570·27-s + 1.21·29-s − 0.417·31-s + 0.176·32-s − 1.18·33-s + 0.746·34-s + 0.685·36-s + 1.53·37-s − 0.683·38-s + 0.591·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.168461208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.168461208\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 80 T + p^{3} T^{2} \) |
| 23 | \( 1 - 112 T + p^{3} T^{2} \) |
| 29 | \( 1 - 190 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 346 T + p^{3} T^{2} \) |
| 41 | \( 1 + 162 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 198 T + p^{3} T^{2} \) |
| 67 | \( 1 - 716 T + p^{3} T^{2} \) |
| 71 | \( 1 - 392 T + p^{3} T^{2} \) |
| 73 | \( 1 - 538 T + p^{3} T^{2} \) |
| 79 | \( 1 - 240 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1072 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1354 T + p^{3} T^{2} \) |
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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
−8.397591200266521026057466993248, −7.945018844646434983698875137092, −7.19181991630308736898043333596, −6.28852008550641375208781918635, −5.32601108786464576100486185696, −4.41962643857109612193908115389, −3.59964584189621243759648336187, −2.85202255844973111807023053002, −2.23804355016947036317753755905, −1.01897678753950536106270279679,
1.01897678753950536106270279679, 2.23804355016947036317753755905, 2.85202255844973111807023053002, 3.59964584189621243759648336187, 4.41962643857109612193908115389, 5.32601108786464576100486185696, 6.28852008550641375208781918635, 7.19181991630308736898043333596, 7.945018844646434983698875137092, 8.397591200266521026057466993248
