L(s) = 1 | + (−0.536 + 0.122i)2-s + (0.855 − 1.77i)3-s + (−1.52 + 0.736i)4-s + (0.610 + 2.67i)5-s + (−0.241 + 1.05i)6-s + (−4.03 − 1.94i)7-s + (1.58 − 1.26i)8-s + (−0.556 − 0.697i)9-s + (−0.654 − 1.35i)10-s + (1.16 + 0.929i)11-s + 3.34i·12-s + (0.906 − 1.13i)13-s + (2.39 + 0.547i)14-s + (5.27 + 1.20i)15-s + (1.41 − 1.78i)16-s − 2.07i·17-s + ⋯ |
L(s) = 1 | + (−0.379 + 0.0865i)2-s + (0.494 − 1.02i)3-s + (−0.764 + 0.368i)4-s + (0.273 + 1.19i)5-s + (−0.0985 + 0.431i)6-s + (−1.52 − 0.733i)7-s + (0.562 − 0.448i)8-s + (−0.185 − 0.232i)9-s + (−0.207 − 0.429i)10-s + (0.351 + 0.280i)11-s + 0.966i·12-s + (0.251 − 0.315i)13-s + (0.640 + 0.146i)14-s + (1.36 + 0.311i)15-s + (0.354 − 0.445i)16-s − 0.502i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.577238 - 0.0569274i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.577238 - 0.0569274i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (3.78 - 3.83i)T \) |
good | 2 | \( 1 + (0.536 - 0.122i)T + (1.80 - 0.867i)T^{2} \) |
| 3 | \( 1 + (-0.855 + 1.77i)T + (-1.87 - 2.34i)T^{2} \) |
| 5 | \( 1 + (-0.610 - 2.67i)T + (-4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (4.03 + 1.94i)T + (4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (-1.16 - 0.929i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.906 + 1.13i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + 2.07iT - 17T^{2} \) |
| 19 | \( 1 + (0.236 + 0.490i)T + (-11.8 + 14.8i)T^{2} \) |
| 23 | \( 1 + (0.382 - 1.67i)T + (-20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (4.54 - 1.03i)T + (27.9 - 13.4i)T^{2} \) |
| 37 | \( 1 + (2.37 - 1.89i)T + (8.23 - 36.0i)T^{2} \) |
| 41 | \( 1 - 0.595iT - 41T^{2} \) |
| 43 | \( 1 + (-0.321 - 0.0733i)T + (38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-8.97 - 7.15i)T + (10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (2.28 + 10.0i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + 4.03T + 59T^{2} \) |
| 61 | \( 1 + (-5.18 + 10.7i)T + (-38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + (5.20 + 6.52i)T + (-14.9 + 65.3i)T^{2} \) |
| 71 | \( 1 + (5.78 - 7.25i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (3.37 + 0.769i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 + (-6.50 + 5.19i)T + (17.5 - 77.0i)T^{2} \) |
| 83 | \( 1 + (0.955 - 0.460i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-4.68 + 1.06i)T + (80.1 - 38.6i)T^{2} \) |
| 97 | \( 1 + (-2.09 - 4.35i)T + (-60.4 + 75.8i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.48673192158270597530198471863, −16.16399155600815304047166333441, −14.32112164011568789258405910800, −13.45265381453446472231804151977, −12.66740352058450840295971734300, −10.46992782139703086448990647580, −9.310468364108848429253135013693, −7.50772521545945450966998841694, −6.70726854227879352784738010540, −3.33608310731647762704031487964,
4.03063395235420135060249814795, 5.73960901976586531278033097929, 8.889036090441205417615811614385, 9.157158320988940573081380684738, 10.24074991179619322850723908248, 12.49016537513863759708879377597, 13.51697837277618606582554393422, 15.00204158920203688087174034810, 16.15547853975778271345648594908, 16.94393094706047321622056833673