| L(s) = 1 | − i·3-s + 4·7-s + 2·9-s − 3i·11-s + 17-s + 7i·19-s − 4i·21-s + 4·23-s − 5i·27-s − 8i·29-s + 4·31-s − 3·33-s − 4i·37-s + 3·41-s + 8i·43-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.51·7-s + 0.666·9-s − 0.904i·11-s + 0.242·17-s + 1.60i·19-s − 0.872i·21-s + 0.834·23-s − 0.962i·27-s − 1.48i·29-s + 0.718·31-s − 0.522·33-s − 0.657i·37-s + 0.468·41-s + 1.21i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.577491927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.577491927\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 7iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 - 4iT - 61T^{2} \) |
| 67 | \( 1 - 9iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - iT - 83T^{2} \) |
| 89 | \( 1 - 13T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−8.282354013491871430601513402853, −7.87018976770035618399839418847, −7.32608330786106190180451574935, −6.19036224680322808017807985565, −5.69515414772576721078801219811, −4.62434033034052160430100928675, −4.02902320048584639973242993028, −2.78346627869216571351558923491, −1.68598405271492766436200814779, −1.00472086545910297191693102776,
1.15715439296928360738055919022, 2.08089019255575196564556233513, 3.23599255119938848603919489012, 4.41192172079942169613829064925, 4.80527064542830406375237522027, 5.33089022352832281038702968692, 6.77759487964778405566023454124, 7.22141428311645768783918999340, 8.006399456635320459510551428068, 8.914815206284655907453304454005
