| L(s) = 1 | + (1.03 + 0.163i)2-s + (−1.27 + 2.50i)3-s + (−0.863 − 0.280i)4-s + (0.639 − 2.14i)5-s + (−1.72 + 2.38i)6-s + (−0.489 + 0.249i)7-s + (−2.70 − 1.37i)8-s + (−2.89 − 3.98i)9-s + (1.01 − 2.10i)10-s + (1.80 − 1.80i)12-s + (0.507 − 3.20i)13-s + (−0.546 + 0.177i)14-s + (4.55 + 4.34i)15-s + (−1.09 − 0.798i)16-s + (−0.885 − 5.59i)17-s + (−2.33 − 4.59i)18-s + ⋯ |
| L(s) = 1 | + (0.729 + 0.115i)2-s + (−0.738 + 1.44i)3-s + (−0.431 − 0.140i)4-s + (0.285 − 0.958i)5-s + (−0.706 + 0.971i)6-s + (−0.185 + 0.0943i)7-s + (−0.957 − 0.487i)8-s + (−0.966 − 1.32i)9-s + (0.319 − 0.666i)10-s + (0.522 − 0.522i)12-s + (0.140 − 0.888i)13-s + (−0.146 + 0.0474i)14-s + (1.17 + 1.12i)15-s + (−0.274 − 0.199i)16-s + (−0.214 − 1.35i)17-s + (−0.551 − 1.08i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.720644 - 0.483292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.720644 - 0.483292i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.639 + 2.14i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-1.03 - 0.163i)T + (1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (1.27 - 2.50i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (0.489 - 0.249i)T + (4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (-0.507 + 3.20i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.885 + 5.59i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.480 - 1.48i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.803 + 0.803i)T + 23iT^{2} \) |
| 29 | \( 1 + (-1.31 + 4.04i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.33 + 0.968i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.434 + 0.852i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-8.50 + 2.76i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (2.55 - 2.55i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.62 + 1.84i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (6.95 + 1.10i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (6.43 + 2.09i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.40 + 7.44i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.62 - 2.62i)T - 67iT^{2} \) |
| 71 | \( 1 + (5.54 + 4.02i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.58 + 8.98i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-3.37 + 2.44i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (11.1 - 1.77i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 - 3.64iT - 89T^{2} \) |
| 97 | \( 1 + (2.60 - 16.4i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−10.33930312310435367929166126438, −9.562635513342669088680721486778, −9.206924944161118500527557035480, −8.037918630862674149279350946597, −6.24375337670988556633267818709, −5.57274375606575348092019307257, −4.86117225877128756548622400280, −4.28301582499309705566862991353, −3.13943626971127741827407307060, −0.42074189165309643689204653325,
1.72245175339817026059749880368, 2.99574003111875387331710391492, 4.26097196889371527740095237296, 5.59607320196275175228398158442, 6.30326041694042219943097959277, 6.91318411038370184526519290530, 7.938074938609413626112613149447, 8.953900164471618042161673108898, 10.17505260490685283010279220663, 11.24527124627054767982783758600
