| L(s) = 1 | − i·3-s + i·5-s + 1.53·7-s − 9-s − 3.80·11-s + 2.39·13-s + 15-s + 2.51i·17-s + 3.13·19-s − 1.53i·21-s + (3.77 + 2.95i)23-s − 25-s + i·27-s − 1.76·29-s + 3.33i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447i·5-s + 0.579·7-s − 0.333·9-s − 1.14·11-s + 0.663·13-s + 0.258·15-s + 0.610i·17-s + 0.719·19-s − 0.334i·21-s + (0.787 + 0.616i)23-s − 0.200·25-s + 0.192i·27-s − 0.328·29-s + 0.599i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.374955240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374955240\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.77 - 2.95i)T \) |
| good | 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 - 2.51iT - 17T^{2} \) |
| 19 | \( 1 - 3.13T + 19T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 - 3.33iT - 31T^{2} \) |
| 37 | \( 1 - 3.84iT - 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 0.753T + 43T^{2} \) |
| 47 | \( 1 + 9.86iT - 47T^{2} \) |
| 53 | \( 1 + 5.02iT - 53T^{2} \) |
| 59 | \( 1 - 11.1iT - 59T^{2} \) |
| 61 | \( 1 - 13.1iT - 61T^{2} \) |
| 67 | \( 1 - 1.33T + 67T^{2} \) |
| 71 | \( 1 + 7.54iT - 71T^{2} \) |
| 73 | \( 1 - 8.62T + 73T^{2} \) |
| 79 | \( 1 + 2.23T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 1.26iT - 97T^{2} \) |
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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
−8.410091205715155511144357672224, −7.47755612433596109334065797429, −7.07962979392916732486040829905, −6.19457544476754935132178883955, −5.41657066249255166975734233459, −4.90638351995933677584072383773, −3.65310583478849474235157299871, −3.02361824262235503795443474448, −2.01822583540854385871570246351, −1.17018691520161033915083192513,
0.37127178415610272700961550016, 1.62285534679290392745693194745, 2.72971200235595798587821249224, 3.47896840475385433512338989697, 4.50406667622931010900618748786, 5.05940512947525297158849953120, 5.56988602985500359543453515741, 6.51683954242196621721714019085, 7.46352279683309659021579159602, 8.076383155469617342304438747431
