| L(s) = 1 | − 1.31·3-s + 1.36·5-s + 4.52i·7-s − 1.27·9-s + i·11-s + 6.15i·13-s − 1.79·15-s − 7.08·17-s + (2.02 + 3.85i)19-s − 5.94i·21-s + 1.57i·23-s − 3.13·25-s + 5.61·27-s + 6.00i·29-s − 1.17·31-s + ⋯ |
| L(s) = 1 | − 0.758·3-s + 0.610·5-s + 1.71i·7-s − 0.424·9-s + 0.301i·11-s + 1.70i·13-s − 0.463·15-s − 1.71·17-s + (0.465 + 0.885i)19-s − 1.29i·21-s + 0.327i·23-s − 0.626·25-s + 1.08·27-s + 1.11i·29-s − 0.210·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5787020153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5787020153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - iT \) |
| 19 | \( 1 + (-2.02 - 3.85i)T \) |
| good | 3 | \( 1 + 1.31T + 3T^{2} \) |
| 5 | \( 1 - 1.36T + 5T^{2} \) |
| 7 | \( 1 - 4.52iT - 7T^{2} \) |
| 13 | \( 1 - 6.15iT - 13T^{2} \) |
| 17 | \( 1 + 7.08T + 17T^{2} \) |
| 23 | \( 1 - 1.57iT - 23T^{2} \) |
| 29 | \( 1 - 6.00iT - 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 10.6iT - 37T^{2} \) |
| 41 | \( 1 + 6.26iT - 41T^{2} \) |
| 43 | \( 1 - 0.714iT - 43T^{2} \) |
| 47 | \( 1 - 1.81iT - 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 5.62T + 59T^{2} \) |
| 61 | \( 1 + 2.63T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 7.28T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 4.65iT - 83T^{2} \) |
| 89 | \( 1 + 7.01iT - 89T^{2} \) |
| 97 | \( 1 - 3.74iT - 97T^{2} \) |
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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
−9.067317306651502221023860448015, −8.659949616223512900254877698906, −7.43491729075805101993899486265, −6.45960963311935624807781148084, −6.10498335779624882334204251232, −5.34927369114353537958584193820, −4.74529703283817973079632408267, −3.58292511985501301400036584963, −2.17385329577872841098741576389, −1.95881772884102000833653843286,
0.21526304362394856885364020014, 1.00691452084489072908325693582, 2.52660556194033249921819114836, 3.42530753625446688954559152987, 4.54870966868151759251808601499, 5.05021689651624779549890176959, 6.16301704449218162174709762263, 6.42246978301902685211857628024, 7.45116174400063398274342173735, 8.055193151219839351266455841864
