| L(s) = 1 | − 2.67·2-s + 3-s + 5.17·4-s − 3.25·5-s − 2.67·6-s + 0.886·7-s − 8.51·8-s + 9-s + 8.72·10-s − 0.591·11-s + 5.17·12-s − 13-s − 2.37·14-s − 3.25·15-s + 12.4·16-s − 1.52·17-s − 2.67·18-s − 3.42·19-s − 16.8·20-s + 0.886·21-s + 1.58·22-s − 9.38·23-s − 8.51·24-s + 5.61·25-s + 2.67·26-s + 27-s + 4.59·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.577·3-s + 2.58·4-s − 1.45·5-s − 1.09·6-s + 0.335·7-s − 3.01·8-s + 0.333·9-s + 2.76·10-s − 0.178·11-s + 1.49·12-s − 0.277·13-s − 0.635·14-s − 0.841·15-s + 3.11·16-s − 0.369·17-s − 0.631·18-s − 0.786·19-s − 3.77·20-s + 0.193·21-s + 0.337·22-s − 1.95·23-s − 1.73·24-s + 1.12·25-s + 0.525·26-s + 0.192·27-s + 0.868·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4071515854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4071515854\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 - 0.886T + 7T^{2} \) |
| 11 | \( 1 + 0.591T + 11T^{2} \) |
| 17 | \( 1 + 1.52T + 17T^{2} \) |
| 19 | \( 1 + 3.42T + 19T^{2} \) |
| 23 | \( 1 + 9.38T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 + 6.01T + 31T^{2} \) |
| 37 | \( 1 - 0.513T + 37T^{2} \) |
| 41 | \( 1 + 3.84T + 41T^{2} \) |
| 43 | \( 1 - 7.11T + 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 2.00T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 9.99T + 83T^{2} \) |
| 89 | \( 1 + 4.29T + 89T^{2} \) |
| 97 | \( 1 - 8.28T + 97T^{2} \) |
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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
−8.505859932613610168563451430325, −7.900752189995852150341919334135, −7.39984632368613909547512835418, −6.79197158628270158125433777317, −5.81249812713215779718124295602, −4.39370149035839212267117271388, −3.65007388834454755767854667972, −2.56362341346418264727064328022, −1.80971683192277198792611333160, −0.45466642578280796255507899662,
0.45466642578280796255507899662, 1.80971683192277198792611333160, 2.56362341346418264727064328022, 3.65007388834454755767854667972, 4.39370149035839212267117271388, 5.81249812713215779718124295602, 6.79197158628270158125433777317, 7.39984632368613909547512835418, 7.900752189995852150341919334135, 8.505859932613610168563451430325
