| L(s) = 1 | − 1.19·3-s − 1.56·5-s + 0.868i·7-s − 1.56·9-s − 3.09i·11-s + 4.74i·13-s + 1.87·15-s − 17-s + (3.07 + 3.09i)19-s − 1.04i·21-s − 3.96i·23-s − 2.56·25-s + 5.47·27-s − 8.45i·29-s + 4.27·31-s + ⋯ |
| L(s) = 1 | − 0.692·3-s − 0.698·5-s + 0.328i·7-s − 0.520·9-s − 0.932i·11-s + 1.31i·13-s + 0.483·15-s − 0.242·17-s + (0.704 + 0.709i)19-s − 0.227i·21-s − 0.825i·23-s − 0.512·25-s + 1.05·27-s − 1.57i·29-s + 0.767·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8122665759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8122665759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.07 - 3.09i)T \) |
| good | 3 | \( 1 + 1.19T + 3T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 0.868iT - 7T^{2} \) |
| 11 | \( 1 + 3.09iT - 11T^{2} \) |
| 13 | \( 1 - 4.74iT - 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 23 | \( 1 + 3.96iT - 23T^{2} \) |
| 29 | \( 1 + 8.45iT - 29T^{2} \) |
| 31 | \( 1 - 4.27T + 31T^{2} \) |
| 37 | \( 1 - 3.70iT - 37T^{2} \) |
| 41 | \( 1 - 3.70iT - 41T^{2} \) |
| 43 | \( 1 + 11.0iT - 43T^{2} \) |
| 47 | \( 1 + 9.27iT - 47T^{2} \) |
| 53 | \( 1 - 1.04iT - 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 0.684T + 61T^{2} \) |
| 67 | \( 1 + 9.74T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 8.12T + 73T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 + 9.65iT - 83T^{2} \) |
| 89 | \( 1 - 5.79iT - 89T^{2} \) |
| 97 | \( 1 + 16.9iT - 97T^{2} \) |
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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
−9.664180806036185917216522523832, −8.613529057436843674651350503287, −8.200688269158820675569382269286, −7.02090358024526402954840089139, −6.20657141607271059967719300576, −5.53060992391914791339245562198, −4.44756047944006206729734416236, −3.56804049861949522846247980347, −2.29062257275408794050755349972, −0.51534631518610585061875270151,
0.914329189675644567367615311382, 2.71717189236734116926929378790, 3.72931275975511294484546323776, 4.87846619818081596990438977566, 5.46888879541730782404069479670, 6.54675510706222987798435653949, 7.43952832400663773769109694706, 7.994443369246924372644814719269, 9.062134387944500245015539417395, 9.933677622433841743084002328125
