| L(s) = 1 | + 2-s + 4-s − 0.592·7-s + 8-s − 11-s − 6.12·13-s − 0.592·14-s + 16-s + 1.40·17-s + 7.52·19-s − 22-s − 5.52·23-s − 6.12·26-s − 0.592·28-s + 1.59·29-s + 3.59·31-s + 32-s + 1.40·34-s − 0.592·37-s + 7.52·38-s + 9.64·41-s + 8.12·43-s − 44-s − 5.52·46-s + 4.59·47-s − 6.64·49-s − 6.12·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.224·7-s + 0.353·8-s − 0.301·11-s − 1.69·13-s − 0.158·14-s + 0.250·16-s + 0.341·17-s + 1.72·19-s − 0.213·22-s − 1.15·23-s − 1.20·26-s − 0.112·28-s + 0.295·29-s + 0.645·31-s + 0.176·32-s + 0.241·34-s − 0.0974·37-s + 1.22·38-s + 1.50·41-s + 1.23·43-s − 0.150·44-s − 0.815·46-s + 0.669·47-s − 0.949·49-s − 0.848·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.810004895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.810004895\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 0.592T + 7T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 - 1.59T + 29T^{2} \) |
| 31 | \( 1 - 3.59T + 31T^{2} \) |
| 37 | \( 1 + 0.592T + 37T^{2} \) |
| 41 | \( 1 - 9.64T + 41T^{2} \) |
| 43 | \( 1 - 8.12T + 43T^{2} \) |
| 47 | \( 1 - 4.59T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 - 4.59T + 59T^{2} \) |
| 61 | \( 1 - 5.18T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 8.34T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 7.81T + 97T^{2} \) |
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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
−7.900376532262649844295006482916, −7.51856942342929174168658261136, −6.85619685349854752836281674573, −5.85352211870621473651994763833, −5.35828176351625443202548005991, −4.59578945893169946927316064864, −3.79619977286420659696126470019, −2.82080592326970019238806310533, −2.26663176994292639643693362082, −0.813733824817284387991896698000,
0.813733824817284387991896698000, 2.26663176994292639643693362082, 2.82080592326970019238806310533, 3.79619977286420659696126470019, 4.59578945893169946927316064864, 5.35828176351625443202548005991, 5.85352211870621473651994763833, 6.85619685349854752836281674573, 7.51856942342929174168658261136, 7.900376532262649844295006482916
