L(s) = 1 | − 1.93·2-s + 1.73·4-s − 0.896·7-s + 0.517·8-s + 4.73·11-s − 2.44·13-s + 1.73·14-s − 4.46·16-s − 0.378·17-s + 2.73·19-s − 9.14·22-s + 1.93·23-s + 4.73·26-s − 1.55·28-s + 6.46·29-s − 2.73·31-s + 7.58·32-s + 0.732·34-s + 4.24·37-s − 5.27·38-s + 4.26·41-s − 9.14·43-s + 8.19·44-s − 3.73·46-s − 4.38·47-s − 6.19·49-s − 4.24·52-s + ⋯ |
L(s) = 1 | − 1.36·2-s + 0.866·4-s − 0.338·7-s + 0.183·8-s + 1.42·11-s − 0.679·13-s + 0.462·14-s − 1.11·16-s − 0.0919·17-s + 0.626·19-s − 1.94·22-s + 0.402·23-s + 0.928·26-s − 0.293·28-s + 1.20·29-s − 0.490·31-s + 1.34·32-s + 0.125·34-s + 0.697·37-s − 0.856·38-s + 0.666·41-s − 1.39·43-s + 1.23·44-s − 0.550·46-s − 0.639·47-s − 0.885·49-s − 0.588·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8210979435\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8210979435\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 7 | \( 1 + 0.896T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + 0.378T + 17T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 + 9.14T + 43T^{2} \) |
| 47 | \( 1 + 4.38T + 47T^{2} \) |
| 53 | \( 1 + 3.86T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 5.13T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 0.535T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 7.39T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−9.262197267237906325299372067551, −8.496347728848632285775690008352, −7.76549669304723326074441101179, −6.87896161009391161588993755632, −6.45489226406458389864199709591, −5.12453945030151772590083408821, −4.20760432611896264583511739759, −3.06975997746268506546654527655, −1.81384019564456910082243449115, −0.76574540539991609095573261270,
0.76574540539991609095573261270, 1.81384019564456910082243449115, 3.06975997746268506546654527655, 4.20760432611896264583511739759, 5.12453945030151772590083408821, 6.45489226406458389864199709591, 6.87896161009391161588993755632, 7.76549669304723326074441101179, 8.496347728848632285775690008352, 9.262197267237906325299372067551