| L(s) = 1 | − 2-s − 2.21·3-s + 4-s − 3.23·5-s + 2.21·6-s + 2.26·7-s − 8-s + 1.88·9-s + 3.23·10-s + 2.49·11-s − 2.21·12-s + 3.08·13-s − 2.26·14-s + 7.14·15-s + 16-s − 3.61·17-s − 1.88·18-s + 5.76·19-s − 3.23·20-s − 5.01·21-s − 2.49·22-s + 5.42·23-s + 2.21·24-s + 5.43·25-s − 3.08·26-s + 2.46·27-s + 2.26·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.27·3-s + 0.5·4-s − 1.44·5-s + 0.902·6-s + 0.857·7-s − 0.353·8-s + 0.628·9-s + 1.02·10-s + 0.753·11-s − 0.638·12-s + 0.856·13-s − 0.606·14-s + 1.84·15-s + 0.250·16-s − 0.876·17-s − 0.444·18-s + 1.32·19-s − 0.722·20-s − 1.09·21-s − 0.532·22-s + 1.13·23-s + 0.451·24-s + 1.08·25-s − 0.605·26-s + 0.474·27-s + 0.428·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7131107636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7131107636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 2.26T + 7T^{2} \) |
| 11 | \( 1 - 2.49T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 23 | \( 1 - 5.42T + 23T^{2} \) |
| 29 | \( 1 + 3.86T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 + 4.62T + 37T^{2} \) |
| 41 | \( 1 - 1.08T + 41T^{2} \) |
| 43 | \( 1 - 5.37T + 43T^{2} \) |
| 47 | \( 1 - 3.52T + 47T^{2} \) |
| 53 | \( 1 + 0.846T + 53T^{2} \) |
| 59 | \( 1 + 2.99T + 59T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 4.18T + 71T^{2} \) |
| 73 | \( 1 + 7.88T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 4.13T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 3.12T + 97T^{2} \) |
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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
−8.510013939529502810651756307545, −7.64229094932542012635555661555, −7.07458133280339896002264429668, −6.39204267992622940761054444212, −5.50731686628185240353169281110, −4.70542457291622412903789554411, −4.00311973983902570778039673492, −2.99947276660656354466843603356, −1.38526840358789067787049335067, −0.64850481401664722546654185914,
0.64850481401664722546654185914, 1.38526840358789067787049335067, 2.99947276660656354466843603356, 4.00311973983902570778039673492, 4.70542457291622412903789554411, 5.50731686628185240353169281110, 6.39204267992622940761054444212, 7.07458133280339896002264429668, 7.64229094932542012635555661555, 8.510013939529502810651756307545
