| L(s) = 1 | + (−2.00 − 1.28i)2-s + (0.142 + 0.989i)3-s + (1.52 + 3.33i)4-s + (0.959 + 0.281i)5-s + (0.988 − 2.16i)6-s + (2.34 − 2.71i)7-s + (0.564 − 3.92i)8-s + (−0.959 + 0.281i)9-s + (−1.55 − 1.79i)10-s + (−3.05 + 1.96i)11-s + (−3.08 + 1.98i)12-s + (1.92 + 2.21i)13-s + (−8.18 + 2.40i)14-s + (−0.142 + 0.989i)15-s + (−1.37 + 1.59i)16-s + (1.83 − 4.01i)17-s + ⋯ |
| L(s) = 1 | + (−1.41 − 0.909i)2-s + (0.0821 + 0.571i)3-s + (0.761 + 1.66i)4-s + (0.429 + 0.125i)5-s + (0.403 − 0.883i)6-s + (0.887 − 1.02i)7-s + (0.199 − 1.38i)8-s + (−0.319 + 0.0939i)9-s + (−0.492 − 0.568i)10-s + (−0.921 + 0.592i)11-s + (−0.890 + 0.572i)12-s + (0.533 + 0.615i)13-s + (−2.18 + 0.642i)14-s + (−0.0367 + 0.255i)15-s + (−0.344 + 0.397i)16-s + (0.444 − 0.973i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.784434 - 0.145639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.784434 - 0.145639i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.40 - 1.90i)T \) |
| good | 2 | \( 1 + (2.00 + 1.28i)T + (0.830 + 1.81i)T^{2} \) |
| 7 | \( 1 + (-2.34 + 2.71i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (3.05 - 1.96i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.92 - 2.21i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.83 + 4.01i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.37 - 5.20i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.477 - 1.04i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.799 + 5.56i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-5.30 + 1.55i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.54 - 0.747i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.454 - 3.16i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 + (-8.00 + 9.23i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.56 - 5.26i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.74 - 12.1i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (8.14 + 5.23i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-11.2 - 7.23i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.37 + 5.20i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (7.71 + 8.90i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (15.2 - 4.48i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.38 + 9.66i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.90 + 1.14i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.17656977329388545632289009272, −10.34128273981638267463021862167, −9.841414839970257189467385586437, −8.958804683500137191773955501161, −7.83680046002128012068249218022, −7.31015447474487463828681872648, −5.41165061612522788350777340652, −4.08285928618729226347111829399, −2.66846759949192608624123541292, −1.27277726356595836751351516992,
1.12333553218650821114474602369, 2.63753029509456717599230607369, 5.30391142693830817963891287741, 5.87717068776497404713184215384, 7.02902727032874324788923458882, 8.110653041278503974723300577976, 8.496438834748516491937982547607, 9.301629436070233193403008939383, 10.55087694896888820527372680600, 11.15812912736044249488574949042
