| L(s) = 1 | + (−1.58 − 0.0830i)2-s + (−1.23 − 1.52i)3-s + (0.513 + 0.0539i)4-s + (1.82 + 2.51i)6-s + (−1.63 − 2.08i)7-s + (2.32 + 0.368i)8-s + (−0.174 + 0.822i)9-s + (−2.03 + 0.433i)11-s + (−0.550 − 0.848i)12-s + (1.94 + 0.992i)13-s + (2.40 + 3.43i)14-s + (−4.66 − 0.990i)16-s + (−1.81 − 4.72i)17-s + (0.345 − 1.28i)18-s + (−0.785 − 7.47i)19-s + ⋯ |
| L(s) = 1 | + (−1.12 − 0.0587i)2-s + (−0.712 − 0.879i)3-s + (0.256 + 0.0269i)4-s + (0.746 + 1.02i)6-s + (−0.616 − 0.787i)7-s + (0.821 + 0.130i)8-s + (−0.0583 + 0.274i)9-s + (−0.614 + 0.130i)11-s + (−0.159 − 0.244i)12-s + (0.540 + 0.275i)13-s + (0.644 + 0.918i)14-s + (−1.16 − 0.247i)16-s + (−0.440 − 1.14i)17-s + (0.0814 − 0.303i)18-s + (−0.180 − 1.71i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0799307 + 0.116029i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0799307 + 0.116029i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + (1.63 + 2.08i)T \) |
| good | 2 | \( 1 + (1.58 + 0.0830i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (1.23 + 1.52i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (2.03 - 0.433i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-1.94 - 0.992i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (1.81 + 4.72i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.785 + 7.47i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.105 + 2.00i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (4.12 - 5.67i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.58 + 5.80i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (2.22 - 1.44i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-8.47 + 2.75i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (0.986 + 0.986i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.45 + 0.941i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-7.17 + 5.81i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (7.04 - 7.81i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (6.37 - 5.73i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (3.06 - 1.17i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-5.41 - 3.93i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.51 - 6.94i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-1.08 - 2.44i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (0.456 - 2.88i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-8.47 - 9.41i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (0.527 + 3.33i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−9.403075783825643473453269878269, −8.936570468173516816504519499290, −7.68897079852106215280553235624, −7.09972558808902439884019917232, −6.55042228880974516629285603804, −5.25006435383932050627200301799, −4.18671875326736287687596521444, −2.53333113752236699405373864865, −1.01903153954517957569549646823, −0.13582149666262337397341275513,
1.84153176451213979834782795966, 3.54082309088402962448570908328, 4.54062146211198845773391572539, 5.69749843804472319687872959673, 6.21700121714931847657893548745, 7.72724170662191709694711277658, 8.306710945116348738670961710934, 9.214764855642592100982125596544, 9.910766460110561791230949493476, 10.54722505785372509185030558467
