L(s) = 1 | + (−2.42 + 4.19i)2-s + (3.09 − 5.36i)3-s + (−7.71 − 13.3i)4-s − 15.2·5-s + (14.9 + 25.9i)6-s + (2.15 + 3.73i)7-s + 35.9·8-s + (−5.69 − 9.87i)9-s + (36.8 − 63.8i)10-s + (12.2 − 21.2i)11-s − 95.5·12-s − 20.8·14-s + (−47.2 + 81.7i)15-s + (−25.2 + 43.8i)16-s + (63.5 + 110. i)17-s + 55.1·18-s + ⋯ |
L(s) = 1 | + (−0.855 + 1.48i)2-s + (0.596 − 1.03i)3-s + (−0.964 − 1.67i)4-s − 1.36·5-s + (1.02 + 1.76i)6-s + (0.116 + 0.201i)7-s + 1.58·8-s + (−0.211 − 0.365i)9-s + (1.16 − 2.02i)10-s + (0.336 − 0.583i)11-s − 2.29·12-s − 0.398·14-s + (−0.812 + 1.40i)15-s + (−0.395 + 0.684i)16-s + (0.907 + 1.57i)17-s + 0.722·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.460430 + 0.686392i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.460430 + 0.686392i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (2.42 - 4.19i)T + (-4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (-3.09 + 5.36i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + 15.2T + 125T^{2} \) |
| 7 | \( 1 + (-2.15 - 3.73i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-12.2 + 21.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (-63.5 - 110. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-25.8 - 44.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (43.6 - 75.6i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (112. - 195. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 108.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (57.8 - 100. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-95.9 + 166. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-61.6 - 106. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 36.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 119.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-402. - 696. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (339. + 587. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-43.7 + 75.7i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-490. - 849. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 - 263.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 321.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-172. + 298. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-241. - 417. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.71410652145989058789147338285, −11.78074498112670909471469979560, −10.35926921941374796958110868037, −8.883336663076914095369207751322, −8.117436499771690708578355111234, −7.71452701347290749586588530133, −6.75704641023544921704236413009, −5.57802633957891116570173379045, −3.66536740685364523214949718793, −1.20759246794816029454293042948,
0.57477301280130435426359043692, 2.74896085645125159519046523020, 3.78646369937656865081338091084, 4.55904323858509427858968893358, 7.38638567744873162212180896730, 8.284898793064608085783597604321, 9.362029829308976452577904333094, 9.879643376479159696277490853501, 10.97998333272784244828141381076, 11.76958006993685481520946760073