L(s) = 1 | + (0.104 + 0.0601i)2-s − 0.582·3-s + (−0.992 − 1.71i)4-s + (1.46 − 0.844i)5-s + (−0.0606 − 0.0350i)6-s − 0.479i·8-s − 2.66·9-s + 0.203·10-s − 0.364i·11-s + (0.578 + 1.00i)12-s + (1.80 − 3.12i)13-s + (−0.851 + 0.491i)15-s + (−1.95 + 3.38i)16-s + (−1.59 − 2.75i)17-s + (−0.277 − 0.160i)18-s − 1.44i·19-s + ⋯ |
L(s) = 1 | + (0.0737 + 0.0425i)2-s − 0.336·3-s + (−0.496 − 0.859i)4-s + (0.653 − 0.377i)5-s + (−0.0247 − 0.0143i)6-s − 0.169i·8-s − 0.886·9-s + 0.0642·10-s − 0.109i·11-s + (0.166 + 0.289i)12-s + (0.499 − 0.866i)13-s + (−0.219 + 0.126i)15-s + (−0.489 + 0.847i)16-s + (−0.386 − 0.669i)17-s + (−0.0653 − 0.0377i)18-s − 0.331i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.255722 - 0.774238i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.255722 - 0.774238i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-1.80 + 3.12i)T \) |
good | 2 | \( 1 + (-0.104 - 0.0601i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + 0.582T + 3T^{2} \) |
| 5 | \( 1 + (-1.46 + 0.844i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 0.364iT - 11T^{2} \) |
| 17 | \( 1 + (1.59 + 2.75i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 1.44iT - 19T^{2} \) |
| 23 | \( 1 + (2.54 - 4.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.09 + 7.09i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.06 + 2.34i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.46 + 3.15i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.04 - 2.91i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.386 - 0.669i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-11.0 + 6.39i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.685 - 1.18i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.10 + 4.68i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 9.02T + 61T^{2} \) |
| 67 | \( 1 + 13.4iT - 67T^{2} \) |
| 71 | \( 1 + (6.13 + 3.54i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.87 - 1.08i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.44 - 5.96i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.567iT - 83T^{2} \) |
| 89 | \( 1 + (-0.986 - 0.569i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.86 - 3.96i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.20086394438805514216347831171, −9.343541464337046782152514790534, −8.782074820827153502575510390330, −7.61559716099771872530647187908, −6.22595750523641561649868270682, −5.62728434136593061598897418578, −5.04634122538000014105213944125, −3.63311802568387077990621226236, −2.01374460532462980798950292803, −0.43665822009123183739497436232,
2.07597237543405583232259579481, 3.33491408374266540928933268194, 4.35958716397527796243396616314, 5.53744488192908569789708593536, 6.40212404750980619377133905724, 7.33664763646722801486946316250, 8.679317304578107361711609026619, 8.837528465479733513914382839333, 10.17488738792584712503252112776, 10.92042356572745127521299243674