L(s) = 1 | − 0.288·3-s + 2.13·5-s + 1.97·7-s − 2.91·9-s − 1.51·11-s + 1.35·13-s − 0.616·15-s + 17-s + 3.95·19-s − 0.567·21-s − 2.03·23-s − 0.420·25-s + 1.70·27-s − 10.0·29-s − 2.29·31-s + 0.436·33-s + 4.21·35-s + 6.48·37-s − 0.389·39-s + 9.92·41-s − 2.66·43-s − 6.24·45-s + 12.6·47-s − 3.11·49-s − 0.288·51-s + 9.27·53-s − 3.23·55-s + ⋯ |
L(s) = 1 | − 0.166·3-s + 0.956·5-s + 0.745·7-s − 0.972·9-s − 0.456·11-s + 0.374·13-s − 0.159·15-s + 0.242·17-s + 0.906·19-s − 0.123·21-s − 0.424·23-s − 0.0841·25-s + 0.328·27-s − 1.86·29-s − 0.411·31-s + 0.0759·33-s + 0.712·35-s + 1.06·37-s − 0.0623·39-s + 1.54·41-s − 0.406·43-s − 0.930·45-s + 1.85·47-s − 0.444·49-s − 0.0403·51-s + 1.27·53-s − 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.322420555\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.322420555\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 0.288T + 3T^{2} \) |
| 5 | \( 1 - 2.13T + 5T^{2} \) |
| 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 2.29T + 31T^{2} \) |
| 37 | \( 1 - 6.48T + 37T^{2} \) |
| 41 | \( 1 - 9.92T + 41T^{2} \) |
| 43 | \( 1 + 2.66T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 9.27T + 53T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 + 5.27T + 73T^{2} \) |
| 79 | \( 1 + 7.76T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 1.97T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−7.70920378929476187650045126915, −7.35416347975422449338505281207, −6.12784801606086017521618771354, −5.68385796990081501137429137496, −5.36472463544220407552370361915, −4.35324419253639613667597382835, −3.46259358212479758148199104850, −2.50245840334697140747824322180, −1.87673259958868079572798615594, −0.76138268875345036390991182594,
0.76138268875345036390991182594, 1.87673259958868079572798615594, 2.50245840334697140747824322180, 3.46259358212479758148199104850, 4.35324419253639613667597382835, 5.36472463544220407552370361915, 5.68385796990081501137429137496, 6.12784801606086017521618771354, 7.35416347975422449338505281207, 7.70920378929476187650045126915