| L(s) = 1 | − 1.93i·3-s − 4.93i·7-s − 0.745·9-s + 0.745·11-s + 1.74i·13-s + 6.10i·17-s − 5.44·19-s − 9.55·21-s − i·23-s − 4.36i·27-s + 1.66·29-s − 1.61·31-s − 1.44i·33-s − 4.34i·37-s + 3.37·39-s + ⋯ |
| L(s) = 1 | − 1.11i·3-s − 1.86i·7-s − 0.248·9-s + 0.224·11-s + 0.484i·13-s + 1.48i·17-s − 1.24·19-s − 2.08·21-s − 0.208i·23-s − 0.839i·27-s + 0.309·29-s − 0.290·31-s − 0.251i·33-s − 0.714i·37-s + 0.541·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4760178689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4760178689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.93iT - 3T^{2} \) |
| 7 | \( 1 + 4.93iT - 7T^{2} \) |
| 11 | \( 1 - 0.745T + 11T^{2} \) |
| 13 | \( 1 - 1.74iT - 13T^{2} \) |
| 17 | \( 1 - 6.10iT - 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 29 | \( 1 - 1.66T + 29T^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 37 | \( 1 + 4.34iT - 37T^{2} \) |
| 41 | \( 1 + 6.95T + 41T^{2} \) |
| 43 | \( 1 - 5.01iT - 43T^{2} \) |
| 47 | \( 1 + 2.68iT - 47T^{2} \) |
| 53 | \( 1 - 13.7iT - 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 13.1iT - 67T^{2} \) |
| 71 | \( 1 + 9.67T + 71T^{2} \) |
| 73 | \( 1 - 5.69iT - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 0.637iT - 83T^{2} \) |
| 89 | \( 1 + 2.72T + 89T^{2} \) |
| 97 | \( 1 + 7.12iT - 97T^{2} \) |
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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
−7.64510109453639092444884024778, −7.19271016416150135325467800361, −6.40705381377134358207069369889, −6.11274483258941312505808345948, −4.47095096408312255648799581633, −4.24226064313106523390457887425, −3.22685803951788473370190911113, −1.83047113678548038974272531737, −1.35162024356850711900845100528, −0.12489895260359317890238115803,
1.79603649955825930895827371337, 2.78180705870331648111531087966, 3.38626602287232560670770665231, 4.57021592440584243123759162702, 5.01153668558753905536227323973, 5.73445496213636607578610493764, 6.45469418068813699109999749919, 7.39211620806571498592970133249, 8.447158371472649501837890978148, 8.852725181981246116838660163508
