L(s) = 1 | + 1.76·2-s + 1.12·4-s − 3.31·5-s + 4.87·7-s − 1.55·8-s − 5.86·10-s + 2.38·11-s − 1.83·13-s + 8.61·14-s − 4.98·16-s + 1.94·19-s − 3.72·20-s + 4.20·22-s + 7.17·23-s + 6.02·25-s − 3.24·26-s + 5.46·28-s − 6.02·29-s + 5.65·31-s − 5.70·32-s − 16.1·35-s + 1.57·37-s + 3.43·38-s + 5.15·40-s + 10.4·41-s + 5.75·43-s + 2.66·44-s + ⋯ |
L(s) = 1 | + 1.24·2-s + 0.560·4-s − 1.48·5-s + 1.84·7-s − 0.549·8-s − 1.85·10-s + 0.717·11-s − 0.509·13-s + 2.30·14-s − 1.24·16-s + 0.445·19-s − 0.831·20-s + 0.896·22-s + 1.49·23-s + 1.20·25-s − 0.636·26-s + 1.03·28-s − 1.11·29-s + 1.01·31-s − 1.00·32-s − 2.73·35-s + 0.258·37-s + 0.556·38-s + 0.815·40-s + 1.63·41-s + 0.878·43-s + 0.402·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.126680363\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.126680363\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 - 1.76T + 2T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 - 7.17T + 23T^{2} \) |
| 29 | \( 1 + 6.02T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 1.57T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 5.75T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 + 9.23T + 53T^{2} \) |
| 59 | \( 1 + 0.726T + 59T^{2} \) |
| 61 | \( 1 - 7.22T + 61T^{2} \) |
| 67 | \( 1 - 5.14T + 67T^{2} \) |
| 71 | \( 1 - 9.34T + 71T^{2} \) |
| 73 | \( 1 + 1.34T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 1.08T + 97T^{2} \) |
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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
−8.755915843672045738337223301004, −7.85908274907782430721994859638, −7.47302288067649819748867085609, −6.51219930810827614141206410126, −5.34975008659740938325764634560, −4.79341658122801858219108989720, −4.23021736437954355064180205222, −3.53802888330195643524042102655, −2.44243810478537382315465210172, −0.969255112279974773142271910569,
0.969255112279974773142271910569, 2.44243810478537382315465210172, 3.53802888330195643524042102655, 4.23021736437954355064180205222, 4.79341658122801858219108989720, 5.34975008659740938325764634560, 6.51219930810827614141206410126, 7.47302288067649819748867085609, 7.85908274907782430721994859638, 8.755915843672045738337223301004