L(s) = 1 | − 1.82·2-s − 1.30·3-s + 1.34·4-s + 2.39·6-s + 2.38·7-s + 1.19·8-s − 1.28·9-s + 4.04·11-s − 1.76·12-s − 0.674·13-s − 4.35·14-s − 4.87·16-s − 0.179·17-s + 2.35·18-s − 2.69·19-s − 3.11·21-s − 7.39·22-s + 0.408·23-s − 1.56·24-s + 1.23·26-s + 5.61·27-s + 3.20·28-s − 6.62·29-s − 0.446·31-s + 6.54·32-s − 5.28·33-s + 0.328·34-s + ⋯ |
L(s) = 1 | − 1.29·2-s − 0.755·3-s + 0.673·4-s + 0.977·6-s + 0.899·7-s + 0.421·8-s − 0.429·9-s + 1.21·11-s − 0.508·12-s − 0.187·13-s − 1.16·14-s − 1.21·16-s − 0.0435·17-s + 0.555·18-s − 0.618·19-s − 0.679·21-s − 1.57·22-s + 0.0852·23-s − 0.318·24-s + 0.242·26-s + 1.07·27-s + 0.606·28-s − 1.22·29-s − 0.0802·31-s + 1.15·32-s − 0.920·33-s + 0.0563·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6847887844\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6847887844\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 3 | \( 1 + 1.30T + 3T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 13 | \( 1 + 0.674T + 13T^{2} \) |
| 17 | \( 1 + 0.179T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 - 0.408T + 23T^{2} \) |
| 29 | \( 1 + 6.62T + 29T^{2} \) |
| 31 | \( 1 + 0.446T + 31T^{2} \) |
| 37 | \( 1 - 6.53T + 37T^{2} \) |
| 41 | \( 1 - 5.81T + 41T^{2} \) |
| 43 | \( 1 - 1.03T + 43T^{2} \) |
| 47 | \( 1 + 9.54T + 47T^{2} \) |
| 53 | \( 1 - 3.34T + 53T^{2} \) |
| 59 | \( 1 + 2.88T + 59T^{2} \) |
| 61 | \( 1 + 6.56T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 5.25T + 71T^{2} \) |
| 73 | \( 1 - 6.90T + 73T^{2} \) |
| 79 | \( 1 - 5.78T + 79T^{2} \) |
| 83 | \( 1 - 17.8T + 83T^{2} \) |
| 89 | \( 1 - 5.55T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.048239684012024344207303619370, −7.66224459709496394468895506552, −6.67041352446614193568950801987, −6.21279837572069989482023084710, −5.21185370986020001955586019385, −4.58123182367127082763001933800, −3.73577492424172337148337159301, −2.32918652908146459301213917916, −1.49486771107642586889449751583, −0.59003110307642740094860691483,
0.59003110307642740094860691483, 1.49486771107642586889449751583, 2.32918652908146459301213917916, 3.73577492424172337148337159301, 4.58123182367127082763001933800, 5.21185370986020001955586019385, 6.21279837572069989482023084710, 6.67041352446614193568950801987, 7.66224459709496394468895506552, 8.048239684012024344207303619370