| L(s) = 1 | + 2·2-s − 2·3-s + 4·4-s − 4·6-s + 8·8-s − 23·9-s − 28·11-s − 8·12-s − 12·13-s + 16·16-s + 64·17-s − 46·18-s + 60·19-s − 56·22-s − 58·23-s − 16·24-s − 24·26-s + 100·27-s + 90·29-s + 128·31-s + 32·32-s + 56·33-s + 128·34-s − 92·36-s + 236·37-s + 120·38-s + 24·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.384·3-s + 1/2·4-s − 0.272·6-s + 0.353·8-s − 0.851·9-s − 0.767·11-s − 0.192·12-s − 0.256·13-s + 1/4·16-s + 0.913·17-s − 0.602·18-s + 0.724·19-s − 0.542·22-s − 0.525·23-s − 0.136·24-s − 0.181·26-s + 0.712·27-s + 0.576·29-s + 0.741·31-s + 0.176·32-s + 0.295·33-s + 0.645·34-s − 0.425·36-s + 1.04·37-s + 0.512·38-s + 0.0985·39-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 64 T + p^{3} T^{2} \) |
| 19 | \( 1 - 60 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 236 T + p^{3} T^{2} \) |
| 41 | \( 1 + 242 T + p^{3} T^{2} \) |
| 43 | \( 1 - 362 T + p^{3} T^{2} \) |
| 47 | \( 1 + 226 T + p^{3} T^{2} \) |
| 53 | \( 1 + 108 T + p^{3} T^{2} \) |
| 59 | \( 1 - 20 T + p^{3} T^{2} \) |
| 61 | \( 1 + 542 T + p^{3} T^{2} \) |
| 67 | \( 1 + 434 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1128 T + p^{3} T^{2} \) |
| 73 | \( 1 + 632 T + p^{3} T^{2} \) |
| 79 | \( 1 + 720 T + p^{3} T^{2} \) |
| 83 | \( 1 - 478 T + p^{3} T^{2} \) |
| 89 | \( 1 - 490 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1456 T + p^{3} T^{2} \) |
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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
−7.971099758056239109820254582762, −7.49616662585999451778460226825, −6.37994167661199817128567159970, −5.77923433160652646150806221144, −5.13219675297520689138979914172, −4.34920326725287752835210588722, −3.14130242534559965020190441836, −2.64399670208461154902627863584, −1.25098539230331182012359554569, 0,
1.25098539230331182012359554569, 2.64399670208461154902627863584, 3.14130242534559965020190441836, 4.34920326725287752835210588722, 5.13219675297520689138979914172, 5.77923433160652646150806221144, 6.37994167661199817128567159970, 7.49616662585999451778460226825, 7.971099758056239109820254582762
