| L(s) = 1 | + (−1.83 − 1.28i)5-s + 3.76i·7-s + 3.54·11-s + 1.00i·13-s + 1.56i·17-s − 7.21·19-s − 6.18i·23-s + (1.71 + 4.69i)25-s − 3·29-s − 5.78·31-s + (4.83 − 6.90i)35-s − 0.851i·37-s + 3.11·41-s − 2.70i·43-s + 9.74i·47-s + ⋯ |
| L(s) = 1 | + (−0.819 − 0.573i)5-s + 1.42i·7-s + 1.06·11-s + 0.278i·13-s + 0.378i·17-s − 1.65·19-s − 1.28i·23-s + (0.342 + 0.939i)25-s − 0.557·29-s − 1.03·31-s + (0.816 − 1.16i)35-s − 0.139i·37-s + 0.487·41-s − 0.412i·43-s + 1.42i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5795405307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5795405307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.83 + 1.28i)T \) |
| good | 7 | \( 1 - 3.76iT - 7T^{2} \) |
| 11 | \( 1 - 3.54T + 11T^{2} \) |
| 13 | \( 1 - 1.00iT - 13T^{2} \) |
| 17 | \( 1 - 1.56iT - 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + 6.18iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 + 0.851iT - 37T^{2} \) |
| 41 | \( 1 - 3.11T + 41T^{2} \) |
| 43 | \( 1 + 2.70iT - 43T^{2} \) |
| 47 | \( 1 - 9.74iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 9.66T + 59T^{2} \) |
| 61 | \( 1 + 8.21T + 61T^{2} \) |
| 67 | \( 1 - 2.91iT - 67T^{2} \) |
| 71 | \( 1 + 7.21T + 71T^{2} \) |
| 73 | \( 1 - 16.7iT - 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 5.33T + 89T^{2} \) |
| 97 | \( 1 - 3.86iT - 97T^{2} \) |
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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
−9.297498564084511667990137142058, −8.903390889840838592184258364490, −8.389823129414413700654542183568, −7.38751380798675500741391387539, −6.35130509092102718497611811679, −5.76402562243765013301168508962, −4.55144425554080612988835867469, −4.02414470112787566896557000967, −2.71850467275802438317073647677, −1.59789616187081932815877777516,
0.22231430375698865140215982564, 1.70438589184194176066146568366, 3.31987516206344561495491372923, 3.90926495770817511687926394863, 4.61959982105536746227064395759, 6.00477718307174292005615548277, 6.90071264385811995094884130493, 7.34130472165422065038852228027, 8.133933215857687479479198463947, 9.078181905266197924175094695359
