| L(s) = 1 | + i·3-s − i·5-s + 0.143·7-s − 9-s + 4.80·11-s + 4.78·13-s + 15-s + 4.86i·17-s + 0.379·19-s + 0.143i·21-s + (1.15 + 4.65i)23-s − 25-s − i·27-s + 8.39·29-s − 2.55i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s + 0.0540·7-s − 0.333·9-s + 1.44·11-s + 1.32·13-s + 0.258·15-s + 1.18i·17-s + 0.0870·19-s + 0.0312i·21-s + (0.240 + 0.970i)23-s − 0.200·25-s − 0.192i·27-s + 1.55·29-s − 0.459i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.423803128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.423803128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-1.15 - 4.65i)T \) |
| good | 7 | \( 1 - 0.143T + 7T^{2} \) |
| 11 | \( 1 - 4.80T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 - 4.86iT - 17T^{2} \) |
| 19 | \( 1 - 0.379T + 19T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 + 2.55iT - 31T^{2} \) |
| 37 | \( 1 + 5.56iT - 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 - 1.03T + 43T^{2} \) |
| 47 | \( 1 - 8.93iT - 47T^{2} \) |
| 53 | \( 1 + 13.3iT - 53T^{2} \) |
| 59 | \( 1 - 6.57iT - 59T^{2} \) |
| 61 | \( 1 - 7.49iT - 61T^{2} \) |
| 67 | \( 1 - 4.96T + 67T^{2} \) |
| 71 | \( 1 + 6.77iT - 71T^{2} \) |
| 73 | \( 1 + 8.04T + 73T^{2} \) |
| 79 | \( 1 + 1.53T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 12.6iT - 89T^{2} \) |
| 97 | \( 1 - 5.67iT - 97T^{2} \) |
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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
−8.461911487794653110788036253299, −7.68007610220928471814480603478, −6.53689381608694893752944707534, −6.17927055135755850030977483923, −5.35735496063580905358824304882, −4.41257153511814223840371717919, −3.85249960334417314776496555277, −3.23658821207426479848924255521, −1.77961797246990955489277272915, −1.03120039176955039430056847091,
0.802249811835910226742798288660, 1.60134127424196461449661110591, 2.77740899619611433565179510622, 3.44571337554512173819710068033, 4.36311625419157854043470283803, 5.19301464675046585189096076003, 6.31539432693125606082497947643, 6.55768824515997623760005243700, 7.13838322968422536184890196444, 8.200649678095559161240651481965
