| L(s) = 1 | − 3-s + 4-s + 0.618·7-s + 9-s − 12-s + 1.61·13-s + 16-s − 1.61·19-s − 0.618·21-s + 25-s − 27-s + 0.618·28-s + 36-s − 0.618·37-s − 1.61·39-s − 0.618·43-s − 48-s − 0.618·49-s + 1.61·52-s + 1.61·57-s + 1.61·61-s + 0.618·63-s + 64-s − 1.61·67-s − 0.618·73-s − 75-s − 1.61·76-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 0.618·7-s + 9-s − 12-s + 1.61·13-s + 16-s − 1.61·19-s − 0.618·21-s + 25-s − 27-s + 0.618·28-s + 36-s − 0.618·37-s − 1.61·39-s − 0.618·43-s − 48-s − 0.618·49-s + 1.61·52-s + 1.61·57-s + 1.61·61-s + 0.618·63-s + 64-s − 1.61·67-s − 0.618·73-s − 75-s − 1.61·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.332686728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.332686728\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.61T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.618T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.618T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.61T + T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.618T + T^{2} \) |
| 79 | \( 1 - 1.61T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.61T + T^{2} \) |
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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
−8.740196774078578981963452134182, −8.217370380470912238602091407302, −7.22824675014273878119193862736, −6.52405731234793272680150454946, −6.09139071185646666948419707896, −5.22810118261683585539336223376, −4.32145630347577758701800795383, −3.40602146691084088686369286300, −2.05229830366043015262863257275, −1.22785703232577407577625024962,
1.22785703232577407577625024962, 2.05229830366043015262863257275, 3.40602146691084088686369286300, 4.32145630347577758701800795383, 5.22810118261683585539336223376, 6.09139071185646666948419707896, 6.52405731234793272680150454946, 7.22824675014273878119193862736, 8.217370380470912238602091407302, 8.740196774078578981963452134182
