| L(s) = 1 | − 0.381·2-s − 3-s − 1.85·4-s − 0.618·5-s + 0.381·6-s − 7-s + 1.47·8-s + 9-s + 0.236·10-s + 1.85·12-s + 3.47·13-s + 0.381·14-s + 0.618·15-s + 3.14·16-s + 3·17-s − 0.381·18-s − 4·19-s + 1.14·20-s + 21-s − 2.76·23-s − 1.47·24-s − 4.61·25-s − 1.32·26-s − 27-s + 1.85·28-s − 3·29-s − 0.236·30-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.577·3-s − 0.927·4-s − 0.276·5-s + 0.155·6-s − 0.377·7-s + 0.520·8-s + 0.333·9-s + 0.0746·10-s + 0.535·12-s + 0.962·13-s + 0.102·14-s + 0.159·15-s + 0.786·16-s + 0.727·17-s − 0.0900·18-s − 0.917·19-s + 0.256·20-s + 0.218·21-s − 0.576·23-s − 0.300·24-s − 0.923·25-s − 0.260·26-s − 0.192·27-s + 0.350·28-s − 0.557·29-s − 0.0430·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6828006914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6828006914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 4.23T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 + 8.09T + 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 7.09T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 6.70T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 + 2.85T + 73T^{2} \) |
| 79 | \( 1 + 0.145T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.958763093816657472040644971478, −8.134522724559763409517473213230, −7.57479334388799130160197045087, −6.46560562518116211451347635065, −5.80142460813670627874827327485, −5.01971804753052512246147262306, −3.98368970320813841273024849397, −3.55794514394223680804351096045, −1.84839343423691775685216957450, −0.56825668943293108956212437309,
0.56825668943293108956212437309, 1.84839343423691775685216957450, 3.55794514394223680804351096045, 3.98368970320813841273024849397, 5.01971804753052512246147262306, 5.80142460813670627874827327485, 6.46560562518116211451347635065, 7.57479334388799130160197045087, 8.134522724559763409517473213230, 8.958763093816657472040644971478
