| L(s) = 1 | − 0.208·2-s − 3-s − 1.95·4-s + 3.22·5-s + 0.208·6-s − 7-s + 0.824·8-s + 9-s − 0.671·10-s + 5.66·11-s + 1.95·12-s − 0.641·13-s + 0.208·14-s − 3.22·15-s + 3.74·16-s − 0.208·18-s − 0.299·19-s − 6.30·20-s + 21-s − 1.18·22-s + 6.01·23-s − 0.824·24-s + 5.37·25-s + 0.133·26-s − 27-s + 1.95·28-s + 7.63·29-s + ⋯ |
| L(s) = 1 | − 0.147·2-s − 0.577·3-s − 0.978·4-s + 1.44·5-s + 0.0851·6-s − 0.377·7-s + 0.291·8-s + 0.333·9-s − 0.212·10-s + 1.70·11-s + 0.564·12-s − 0.178·13-s + 0.0557·14-s − 0.831·15-s + 0.935·16-s − 0.0491·18-s − 0.0687·19-s − 1.40·20-s + 0.218·21-s − 0.251·22-s + 1.25·23-s − 0.168·24-s + 1.07·25-s + 0.0262·26-s − 0.192·27-s + 0.369·28-s + 1.41·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.877404250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877404250\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.208T + 2T^{2} \) |
| 5 | \( 1 - 3.22T + 5T^{2} \) |
| 11 | \( 1 - 5.66T + 11T^{2} \) |
| 13 | \( 1 + 0.641T + 13T^{2} \) |
| 19 | \( 1 + 0.299T + 19T^{2} \) |
| 23 | \( 1 - 6.01T + 23T^{2} \) |
| 29 | \( 1 - 7.63T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 + 3.76T + 41T^{2} \) |
| 43 | \( 1 + 4.08T + 43T^{2} \) |
| 47 | \( 1 + 2.80T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 + 2.95T + 59T^{2} \) |
| 61 | \( 1 - 3.36T + 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 - 8.78T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 1.38T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.391404525798331131374550213240, −7.06097962800938820657766974337, −6.55907438376364727109839059165, −5.99019170799899709199837029314, −5.19564498438453427136295118275, −4.61356795031147408741402591322, −3.76975418409712001720172549580, −2.75718350751196659181027092164, −1.50331988805979872274544126140, −0.859253747674413634672363499005,
0.859253747674413634672363499005, 1.50331988805979872274544126140, 2.75718350751196659181027092164, 3.76975418409712001720172549580, 4.61356795031147408741402591322, 5.19564498438453427136295118275, 5.99019170799899709199837029314, 6.55907438376364727109839059165, 7.06097962800938820657766974337, 8.391404525798331131374550213240
