| L(s) = 1 | − 0.732i·3-s − 2.73i·5-s − 7-s + 2.46·9-s + 1.46i·11-s + 2.73i·13-s − 2·15-s + 7.46·17-s + 6.19i·19-s + 0.732i·21-s + 8.92·23-s − 2.46·25-s − 4i·27-s + 3.46i·29-s − 2.53·31-s + ⋯ |
| L(s) = 1 | − 0.422i·3-s − 1.22i·5-s − 0.377·7-s + 0.821·9-s + 0.441i·11-s + 0.757i·13-s − 0.516·15-s + 1.81·17-s + 1.42i·19-s + 0.159i·21-s + 1.86·23-s − 0.492·25-s − 0.769i·27-s + 0.643i·29-s − 0.455·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.950494694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.950494694\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 0.732iT - 3T^{2} \) |
| 5 | \( 1 + 2.73iT - 5T^{2} \) |
| 11 | \( 1 - 1.46iT - 11T^{2} \) |
| 13 | \( 1 - 2.73iT - 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 - 6.19iT - 19T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 4.53iT - 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 2.53iT - 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 6.19iT - 59T^{2} \) |
| 61 | \( 1 + 11.1iT - 61T^{2} \) |
| 67 | \( 1 - 14.9iT - 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 8.92T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 7.26iT - 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 0.535T + 97T^{2} \) |
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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
−9.201949957790632627247871705762, −8.429214239042890345687998083535, −7.50193311687930774139188970721, −7.01344055170672613366470297525, −5.86303884873319405057522345945, −5.12889483503851545995327845737, −4.25659509208503848029010951048, −3.32682980485732716543354319115, −1.74138715335183066144175216354, −1.03278901306099974989180666306,
1.04232975951648635619868290700, 3.00504912333611367271811465406, 3.06544747139649413634213611253, 4.36798922581510311801843612582, 5.36657832177100006348826173675, 6.19955760400706632589268625566, 7.23561477128058591308078673642, 7.42896733827091996847078191935, 8.717637930636329088632503813780, 9.542491913487184247991952449568
