| L(s) = 1 | + (−25.8 − 18.8i)2-s + (2.50 − 7.69i)3-s + (316. + 973. i)4-s + (−1.00e4 + 7.27e3i)5-s + (−209. + 152. i)6-s + (−8.89e3 − 2.73e4i)7-s + (1.01e4 − 3.11e4i)8-s + (1.43e5 + 1.04e5i)9-s + 3.96e5·10-s + (−4.79e5 + 2.35e5i)11-s + 8.28e3·12-s + (1.17e5 + 8.55e4i)13-s + (−2.84e5 + 8.76e5i)14-s + (3.09e4 + 9.52e4i)15-s + (−8.48e5 + 6.16e5i)16-s + (5.91e6 − 4.30e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.00594 − 0.0182i)3-s + (0.154 + 0.475i)4-s + (−1.43 + 1.04i)5-s + (−0.0110 + 0.00799i)6-s + (−0.200 − 0.615i)7-s + (0.109 − 0.336i)8-s + (0.808 + 0.587i)9-s + 1.25·10-s + (−0.897 + 0.441i)11-s + 0.00961·12-s + (0.0879 + 0.0639i)13-s + (−0.141 + 0.435i)14-s + (0.0105 + 0.0323i)15-s + (−0.202 + 0.146i)16-s + (1.01 − 0.734i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.535066 - 0.452438i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.535066 - 0.452438i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (25.8 + 18.8i)T \) |
| 11 | \( 1 + (4.79e5 - 2.35e5i)T \) |
| good | 3 | \( 1 + (-2.50 + 7.69i)T + (-1.43e5 - 1.04e5i)T^{2} \) |
| 5 | \( 1 + (1.00e4 - 7.27e3i)T + (1.50e7 - 4.64e7i)T^{2} \) |
| 7 | \( 1 + (8.89e3 + 2.73e4i)T + (-1.59e9 + 1.16e9i)T^{2} \) |
| 13 | \( 1 + (-1.17e5 - 8.55e4i)T + (5.53e11 + 1.70e12i)T^{2} \) |
| 17 | \( 1 + (-5.91e6 + 4.30e6i)T + (1.05e13 - 3.25e13i)T^{2} \) |
| 19 | \( 1 + (-4.87e6 + 1.50e7i)T + (-9.42e13 - 6.84e13i)T^{2} \) |
| 23 | \( 1 + 3.40e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (4.40e7 + 1.35e8i)T + (-9.87e15 + 7.17e15i)T^{2} \) |
| 31 | \( 1 + (-1.17e8 - 8.54e7i)T + (7.85e15 + 2.41e16i)T^{2} \) |
| 37 | \( 1 + (-1.83e7 - 5.63e7i)T + (-1.43e17 + 1.04e17i)T^{2} \) |
| 41 | \( 1 + (-2.58e8 + 7.95e8i)T + (-4.45e17 - 3.23e17i)T^{2} \) |
| 43 | \( 1 - 9.13e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (7.30e8 - 2.24e9i)T + (-2.00e18 - 1.45e18i)T^{2} \) |
| 53 | \( 1 + (-3.21e9 - 2.33e9i)T + (2.86e18 + 8.81e18i)T^{2} \) |
| 59 | \( 1 + (-1.16e8 - 3.58e8i)T + (-2.43e19 + 1.77e19i)T^{2} \) |
| 61 | \( 1 + (-5.49e9 + 3.99e9i)T + (1.34e19 - 4.13e19i)T^{2} \) |
| 67 | \( 1 + 1.62e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-1.22e10 + 8.92e9i)T + (7.14e19 - 2.19e20i)T^{2} \) |
| 73 | \( 1 + (6.22e9 + 1.91e10i)T + (-2.53e20 + 1.84e20i)T^{2} \) |
| 79 | \( 1 + (1.69e10 + 1.23e10i)T + (2.31e20 + 7.11e20i)T^{2} \) |
| 83 | \( 1 + (2.06e10 - 1.50e10i)T + (3.97e20 - 1.22e21i)T^{2} \) |
| 89 | \( 1 - 3.02e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (5.94e10 + 4.31e10i)T + (2.21e21 + 6.80e21i)T^{2} \) |
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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
−15.47808897491971028637604552321, −13.72160141488294863363165042220, −12.13687952279657081520018340205, −10.94582231691863306628376486493, −10.00234815391087159326926231487, −7.76022824664155170114280630525, −7.23011736585939716725896960343, −4.25258641339858302511668161053, −2.75709385405113435183061091569, −0.41532745678552956012229838618,
1.05446141579936557394271725467, 3.81694392017181645305792503198, 5.61361954263600489704995136786, 7.67059228543021695470830348025, 8.513720259295068093146523322387, 10.05312189681811865312808543833, 11.88095548272073024525069977561, 12.75119149274106493705219090172, 14.89993873717018709960083690415, 15.95171035156787551199390217047
