| L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s + 2.60i·7-s − i·8-s + 9-s − 10-s + 12-s − 3.60·13-s − 2.60·14-s − i·15-s + 16-s − 2.60·17-s + i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s + 0.984i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s − 1.00·13-s − 0.696·14-s − 0.258i·15-s + 0.250·16-s − 0.631·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(-0.621359i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.621359i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + 3.60T \) |
| good | 7 | \( 1 - 2.60iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 - 2.60iT - 19T^{2} \) |
| 23 | \( 1 + 8.60T + 23T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 - 5.21iT - 37T^{2} \) |
| 41 | \( 1 + 11.2iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 5.21iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 5.21iT - 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 5.21iT - 71T^{2} \) |
| 73 | \( 1 + 8.60iT - 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 17.2iT - 83T^{2} \) |
| 89 | \( 1 - 0.788iT - 89T^{2} \) |
| 97 | \( 1 - 8.60iT - 97T^{2} \) |
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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
−12.08933501134620502520807611218, −10.77228316551657251216793525247, −9.910268444603219405353479661436, −8.979227948683916513209908153133, −7.921734429351138403206818904362, −6.96958288781816199857496169008, −6.01134239533421212758117675398, −5.28244379914479056395443846226, −4.02711500343632160153733806308, −2.30692486753724618079884139079,
0.42435673838783386560341274055, 2.16506396219071637825830815672, 3.96000082420657021028174005107, 4.67881252431924868672926182178, 5.88514292279929572146331415696, 7.15770020134942168082494972496, 8.061410447807527070752279740101, 9.391896866180969034208771147143, 10.02697477206069048842556033555, 10.94247011434234298188878366095
