| L(s) = 1 | − 0.322·2-s + 1.07·3-s − 1.89·4-s − 0.347·6-s − 7-s + 1.25·8-s − 1.83·9-s + 1.93·11-s − 2.04·12-s − 1.66·13-s + 0.322·14-s + 3.38·16-s + 2.35·17-s + 0.592·18-s − 1.86·19-s − 1.07·21-s − 0.624·22-s − 23-s + 1.35·24-s + 0.538·26-s − 5.21·27-s + 1.89·28-s + 6.67·29-s + 3.84·31-s − 3.60·32-s + 2.08·33-s − 0.758·34-s + ⋯ |
| L(s) = 1 | − 0.227·2-s + 0.622·3-s − 0.948·4-s − 0.141·6-s − 0.377·7-s + 0.443·8-s − 0.612·9-s + 0.584·11-s − 0.590·12-s − 0.463·13-s + 0.0861·14-s + 0.846·16-s + 0.570·17-s + 0.139·18-s − 0.427·19-s − 0.235·21-s − 0.133·22-s − 0.208·23-s + 0.276·24-s + 0.105·26-s − 1.00·27-s + 0.358·28-s + 1.23·29-s + 0.689·31-s − 0.636·32-s + 0.363·33-s − 0.130·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.322T + 2T^{2} \) |
| 3 | \( 1 - 1.07T + 3T^{2} \) |
| 11 | \( 1 - 1.93T + 11T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 + 1.86T + 19T^{2} \) |
| 29 | \( 1 - 6.67T + 29T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 + 0.829T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 0.466T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 - 1.48T + 61T^{2} \) |
| 67 | \( 1 - 3.24T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 0.0689T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 8.54T + 89T^{2} \) |
| 97 | \( 1 - 1.35T + 97T^{2} \) |
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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
−8.180605140458491191173156019808, −7.67241481619387874685203637265, −6.58882909613990380243880188166, −5.88792770492535783182279667303, −4.95010511923852252790431744928, −4.22467481118534992452188911190, −3.36786340745026845886797031620, −2.63545933301355185771044035935, −1.30726674844136822256674927873, 0,
1.30726674844136822256674927873, 2.63545933301355185771044035935, 3.36786340745026845886797031620, 4.22467481118534992452188911190, 4.95010511923852252790431744928, 5.88792770492535783182279667303, 6.58882909613990380243880188166, 7.67241481619387874685203637265, 8.180605140458491191173156019808
