| L(s) = 1 | − 0.599·2-s − 1.64·4-s − 0.261·5-s − 0.444·7-s + 2.18·8-s + 0.156·10-s + 3.27·11-s + 4.82·13-s + 0.266·14-s + 1.97·16-s − 0.628·17-s + 6.37·19-s + 0.428·20-s − 1.96·22-s + 6.91·23-s − 4.93·25-s − 2.89·26-s + 0.728·28-s + 3.92·29-s + 3.84·31-s − 5.54·32-s + 0.376·34-s + 0.115·35-s − 7.74·37-s − 3.82·38-s − 0.570·40-s − 12.1·41-s + ⋯ |
| L(s) = 1 | − 0.423·2-s − 0.820·4-s − 0.116·5-s − 0.167·7-s + 0.771·8-s + 0.0495·10-s + 0.986·11-s + 1.33·13-s + 0.0711·14-s + 0.493·16-s − 0.152·17-s + 1.46·19-s + 0.0958·20-s − 0.418·22-s + 1.44·23-s − 0.986·25-s − 0.567·26-s + 0.137·28-s + 0.729·29-s + 0.690·31-s − 0.980·32-s + 0.0646·34-s + 0.0196·35-s − 1.27·37-s − 0.620·38-s − 0.0901·40-s − 1.89·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.436290826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436290826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 0.599T + 2T^{2} \) |
| 5 | \( 1 + 0.261T + 5T^{2} \) |
| 7 | \( 1 + 0.444T + 7T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 0.628T + 17T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 - 3.92T + 29T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 + 7.74T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 3.58T + 43T^{2} \) |
| 47 | \( 1 - 6.18T + 47T^{2} \) |
| 53 | \( 1 - 2.50T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 - 2.21T + 61T^{2} \) |
| 67 | \( 1 - 3.22T + 67T^{2} \) |
| 71 | \( 1 - 9.85T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 9.04T + 79T^{2} \) |
| 83 | \( 1 - 5.99T + 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 97 | \( 1 + 5.55T + 97T^{2} \) |
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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
−8.650935779779487104087958100303, −7.86613833317736159716804616428, −7.03674621532660816198545079913, −6.32298320020189987176125128176, −5.39617526168341310330037976232, −4.68657265336636566882000672883, −3.72162085195101197977010499083, −3.24820637914412195116916231083, −1.56906708625116468789290067933, −0.818752187425564275421096348085,
0.818752187425564275421096348085, 1.56906708625116468789290067933, 3.24820637914412195116916231083, 3.72162085195101197977010499083, 4.68657265336636566882000672883, 5.39617526168341310330037976232, 6.32298320020189987176125128176, 7.03674621532660816198545079913, 7.86613833317736159716804616428, 8.650935779779487104087958100303
