L(s) = 1 | + 367. i·2-s + (9.13e4 − 4.60e4i)3-s + 1.96e6·4-s + 3.23e7·5-s + (1.69e7 + 3.35e7i)6-s + (−4.79e8 − 5.73e8i)7-s + 1.49e9i·8-s + (6.22e9 − 8.40e9i)9-s + 1.18e10i·10-s + 1.31e11i·11-s + (1.79e11 − 9.03e10i)12-s − 6.69e11i·13-s + (2.10e11 − 1.75e11i)14-s + (2.95e12 − 1.48e12i)15-s + 3.56e12·16-s + 9.54e12·17-s + ⋯ |
L(s) = 1 | + 0.253i·2-s + (0.892 − 0.450i)3-s + 0.935·4-s + 1.48·5-s + (0.114 + 0.226i)6-s + (−0.641 − 0.767i)7-s + 0.490i·8-s + (0.594 − 0.803i)9-s + 0.375i·10-s + 1.52i·11-s + (0.835 − 0.421i)12-s − 1.34i·13-s + (0.194 − 0.162i)14-s + (1.32 − 0.666i)15-s + 0.811·16-s + 1.14·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(5.029585890\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.029585890\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-9.13e4 + 4.60e4i)T \) |
| 7 | \( 1 + (4.79e8 + 5.73e8i)T \) |
good | 2 | \( 1 - 367. iT - 2.09e6T^{2} \) |
| 5 | \( 1 - 3.23e7T + 4.76e14T^{2} \) |
| 11 | \( 1 - 1.31e11iT - 7.40e21T^{2} \) |
| 13 | \( 1 + 6.69e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 - 9.54e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.11e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 - 2.12e14iT - 3.94e28T^{2} \) |
| 29 | \( 1 - 1.26e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 + 5.66e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 + 4.08e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 2.27e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.15e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 1.81e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 7.35e17iT - 1.62e36T^{2} \) |
| 59 | \( 1 - 2.94e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.08e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 + 6.78e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.06e19iT - 7.52e38T^{2} \) |
| 73 | \( 1 - 4.03e19iT - 1.34e39T^{2} \) |
| 79 | \( 1 - 9.41e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.74e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.43e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 8.62e20iT - 5.27e41T^{2} \) |
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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
−13.34522537449287031186697939828, −12.46814087965536243994943161049, −10.26660486600820083715665778232, −9.645170808350891579239114986663, −7.68669538982983945278571317657, −6.85287826311048579595815482836, −5.56700955649146451985843817170, −3.28118011315050020021220367628, −2.17622385748803838257409677250, −1.20138009126360946893523276377,
1.48283537260570273140306996586, 2.46604871909020172558541621733, 3.38023206354061251101800851253, 5.60690987407345271760885342308, 6.61407977986878479362446660670, 8.585303693737112865794312444438, 9.634600770653356348789824479419, 10.61731291642779476954639374311, 12.25470259830762909140619792682, 13.68995550272825648584775282421